Left Termination of the query pattern qs_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇔)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
      ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔)
        ↳27 TRUE
      ↳28 PiDP
        ↳29 PiDPToQDPProof (⇐)
        ↳30 QDP
  ↳31 PrologToPiTRSProof (⇐)
    ↳32 PiTRS
    ↳33 DependencyPairsProof (⇔)
    ↳34 PiDP
    ↳35 DependencyGraphProof (⇔)
    ↳36 AND
      ↳37 PiDP
        ↳38 UsableRulesProof (⇔)
        ↳39 PiDP
        ↳40 PiDPToQDPProof (⇐)
        ↳41 QDP
        ↳42 QDPSizeChangeProof (⇔)
        ↳43 TRUE
      ↳44 PiDP
        ↳45 UsableRulesProof (⇔)
        ↳46 PiDP
        ↳47 PiDPToQDPProof (⇔)
        ↳48 QDP
        ↳49 QDPSizeChangeProof (⇔)
        ↳50 TRUE
      ↳51 PiDP
        ↳52 UsableRulesProof (⇔)
        ↳53 PiDP
        ↳54 PiDPToQDPProof (⇐)
        ↳55 QDP
        ↳56 QDPSizeChangeProof (⇔)
        ↳57 TRUE
      ↳58 PiDP
        ↳59 PiDPToQDPProof (⇐)
        ↳60 QDP
        ↳61 QDPOrderProof (⇔)
        ↳62 QDP
        ↳63 DependencyGraphProof (⇔)
        ↳64 TRUE

(0) Obligation:

Clauses:

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

Queries:

qs(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

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

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)

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

(4) Obligation:

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

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

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(13) TRUE

(14) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(20) TRUE

(21) Obligation:

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

U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U5_GGAA(X, Y, Xs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs)) → PART_IN_GGAA(X, Xs)

The TRS R consists of the following rules:

less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

less_in_gg(x0, x1)
U9_gg(x0, x1, x2)

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, less_in_gg(X, Y))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

  • PART_IN_GGAA(X, .(Y, Xs)) → PART_IN_GGAA(X, Xs)
    The graph contains the following edges 1 >= 1, 2 > 2

  • U5_GGAA(X, Y, Xs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs)
    The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2

(27) TRUE

(28) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x2, x3, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)

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

(29) PiDPToQDPProof (SOUND transformation)

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

(30) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ga([]) → qs_out_ga([], [])
qs_in_ga(.(X, Xs)) → U1_ga(X, Xs, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
part_in_ggaa(X1, []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys) → U8_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
U8_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The set Q consists of the following terms:

qs_in_ga(x0)
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0, x1, x2)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1, x2, x3)
U6_ggaa(x0, x1, x2, x3)
U1_ga(x0, x1, x2)
U2_ga(x0, x1, x2, x3)
U3_ga(x0, x1, x2, x3)
app_in_gga(x0, x1)
U8_gga(x0, x1, x2, x3)
U4_ga(x0, x1, x2)

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

(31) PrologToPiTRSProof (SOUND transformation)

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

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(32) Obligation:

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

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)

(33) DependencyPairsProof (EQUIVALENT transformation)

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x2, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x2, x6)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x5)

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

(34) Obligation:

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

QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x3)
U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x2, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x2, x6)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x1, x5)

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

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.

(36) Complex Obligation (AND)

(37) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

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

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

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

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

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

(42) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(43) TRUE

(44) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(45) UsableRulesProof (EQUIVALENT transformation)

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

(46) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(47) PiDPToQDPProof (EQUIVALENT transformation)

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

(48) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(49) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(50) TRUE

(51) Obligation:

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

U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(52) UsableRulesProof (EQUIVALENT transformation)

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

(53) Obligation:

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

U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

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

(54) PiDPToQDPProof (SOUND transformation)

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

(55) Obligation:

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

U5_GGAA(X, Y, Xs, less_out_gg) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs)) → PART_IN_GGAA(X, Xs)

The TRS R consists of the following rules:

less_in_gg(0, s(X2)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
U9_gg(x0)

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

(56) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, less_in_gg(X, Y))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

  • PART_IN_GGAA(X, .(Y, Xs)) → PART_IN_GGAA(X, Xs)
    The graph contains the following edges 1 >= 1, 2 > 2

  • U5_GGAA(X, Y, Xs, less_out_gg) → PART_IN_GGAA(X, Xs)
    The graph contains the following edges 1 >= 1, 3 >= 2

(57) TRUE

(58) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg(0, s(X2))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X1, [], [], []) → part_out_ggaa(X1, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2)  =  qs_in_ga(x1)
[]  =  []
qs_out_ga(x1, x2)  =  qs_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x1, x5)
QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)

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

(59) PiDPToQDPProof (SOUND transformation)

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

(60) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ga([]) → qs_out_ga([])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X1, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U8_gga(X, app_in_gga(Xs, Ys))
U8_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)

The set Q consists of the following terms:

qs_in_ga(x0)
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(0) = 1   
POL(QS_IN_GA(x1)) = x1   
POL(U1_GA(x1, x2)) = x2   
POL(U1_ga(x1, x2)) = 0   
POL(U2_GA(x1, x2, x3)) = x2   
POL(U2_ga(x1, x2, x3)) = 0   
POL(U3_ga(x1, x2, x3)) = 0   
POL(U4_ga(x1)) = 0   
POL(U5_ggaa(x1, x2, x3, x4)) = 1 + x2 + x3   
POL(U6_ggaa(x1, x2)) = 1 + x1 + x2   
POL(U7_ggaa(x1, x2)) = 1 + x1 + x2   
POL(U8_gga(x1, x2)) = 1 + x1 + x2   
POL(U9_gg(x1)) = 1 + x1   
POL([]) = 0   
POL(app_in_gga(x1, x2)) = x1 + x2   
POL(app_out_gga(x1)) = x1   
POL(less_in_gg(x1, x2)) = x1   
POL(less_out_gg) = 1   
POL(part_in_ggaa(x1, x2)) = x2   
POL(part_out_ggaa(x1, x2)) = x1 + x2   
POL(qs_in_ga(x1)) = 0   
POL(qs_out_ga(x1)) = 0   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X1, []) → part_out_ggaa([], [])
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))

(62) Obligation:

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

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

The TRS R consists of the following rules:

qs_in_ga([]) → qs_out_ga([])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
less_in_gg(0, s(X2)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X1, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U8_gga(X, app_in_gga(Xs, Ys))
U8_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)

The set Q consists of the following terms:

qs_in_ga(x0)
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)

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

(63) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(64) TRUE