Left Termination of the query pattern ss_in_2(g, g) 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 MRRProof (⇔)
        ↳20 QDP
        ↳21 DependencyGraphProof (⇔)
        ↳22 TRUE
      ↳23 PiDP
        ↳24 UsableRulesProof (⇔)
        ↳25 PiDP
        ↳26 PiDPToQDPProof (⇐)
        ↳27 QDP
        ↳28 QDPSizeChangeProof (⇔)
        ↳29 TRUE
      ↳30 PiDP
        ↳31 UsableRulesProof (⇔)
        ↳32 PiDP
        ↳33 PiDPToQDPProof (⇐)
        ↳34 QDP
        ↳35 QDPSizeChangeProof (⇔)
        ↳36 TRUE
      ↳37 PiDP
        ↳38 UsableRulesProof (⇔)
        ↳39 PiDP
  ↳40 PrologToPiTRSProof (⇐)
    ↳41 PiTRS
    ↳42 DependencyPairsProof (⇔)
    ↳43 PiDP
    ↳44 DependencyGraphProof (⇔)
    ↳45 AND
      ↳46 PiDP
        ↳47 UsableRulesProof (⇔)
        ↳48 PiDP
        ↳49 PiDPToQDPProof (⇔)
        ↳50 QDP
        ↳51 QDPSizeChangeProof (⇔)
        ↳52 TRUE
      ↳53 PiDP
        ↳54 UsableRulesProof (⇔)
        ↳55 PiDP
        ↳56 PiDPToQDPProof (⇐)
        ↳57 QDP
        ↳58 UsableRulesReductionPairsProof (⇔)
        ↳59 QDP
        ↳60 MRRProof (⇔)
        ↳61 QDP
        ↳62 DependencyGraphProof (⇔)
        ↳63 TRUE
      ↳64 PiDP
        ↳65 UsableRulesProof (⇔)
        ↳66 PiDP
        ↳67 PiDPToQDPProof (⇐)
        ↳68 QDP
        ↳69 QDPSizeChangeProof (⇔)
        ↳70 TRUE
      ↳71 PiDP
        ↳72 UsableRulesProof (⇔)
        ↳73 PiDP
        ↳74 PiDPToQDPProof (⇐)
        ↳75 QDP
        ↳76 QDPSizeChangeProof (⇔)
        ↳77 TRUE
      ↳78 PiDP
        ↳79 UsableRulesProof (⇔)
        ↳80 PiDP
        ↳81 PiDPToQDPProof (⇐)
        ↳82 QDP
        ↳83 QDPSizeChangeProof (⇔)
        ↳84 TRUE

(0) Obligation:

Clauses:

ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)).
perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
ordered([]).
ordered(.(X1, [])).
ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

ss(g,g).

(1) PrologToPiTRSProof (SOUND transformation)

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

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)

(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:

SS_IN_GG(Xs, Ys) → U1_GG(Xs, Ys, perm_in_gg(Xs, Ys))
SS_IN_GG(Xs, Ys) → PERM_IN_GG(Xs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GG(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_GG(Xs, X, Ys, perm_in_gg(Zs, Ys))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_GG(Xs, Ys, ordered_in_g(Ys))
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(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)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)
SS_IN_GG(x1, x2)  =  SS_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x1, x4, x5)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)

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

(4) Obligation:

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

SS_IN_GG(Xs, Ys) → U1_GG(Xs, Ys, perm_in_gg(Xs, Ys))
SS_IN_GG(Xs, Ys) → PERM_IN_GG(Xs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GG(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_GG(Xs, X, Ys, perm_in_gg(Zs, Ys))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_GG(Xs, Ys, ordered_in_g(Ys))
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(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)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)
SS_IN_GG(x1, x2)  =  SS_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x1, x4, x5)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x1, x2, x3, x4)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 12 less nodes.

(6) Complex Obligation (AND)

(7) 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:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(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:

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

(10) PiDPToQDPProof (EQUIVALENT 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:

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.

(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:

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

U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)
ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)

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:

U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))

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))

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:

U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))

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.

(19) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))


Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + 2·x1 + x2   
POL(0) = 0   
POL(ORDERED_IN_G(x1)) = 2·x1   
POL(U7_G(x1, x2, x3, x4)) = 2 + x1 + 2·x2 + 2·x3 + x4   
POL(U9_gg(x1, x2, x3)) = 2·x1 + x2 + x3   
POL(less_in_gg(x1, x2)) = 2·x1 + x2   
POL(less_out_gg(x1, x2)) = x1 + x2   
POL(s(x1)) = 2·x1   

(20) Obligation:

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

U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, 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.

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(22) TRUE

(23) 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:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) 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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) 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.

(28) 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

(29) TRUE

(30) Obligation:

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

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

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

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

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

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

(35) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(36) TRUE

(37) Obligation:

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

U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x1, x2, x3, x4)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x2, x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g(x1)
U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
ss_out_gg(x1, x2)  =  ss_out_gg(x1, x2)
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)

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:

U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x4, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x3, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)

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

(40) PrologToPiTRSProof (SOUND transformation)

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

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(41) Obligation:

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

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg

(42) 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:

SS_IN_GG(Xs, Ys) → U1_GG(Xs, Ys, perm_in_gg(Xs, Ys))
SS_IN_GG(Xs, Ys) → PERM_IN_GG(Xs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GG(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_GG(Xs, X, Ys, perm_in_gg(Zs, Ys))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_GG(Xs, Ys, ordered_in_g(Ys))
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(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)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg
SS_IN_GG(x1, x2)  =  SS_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x2, x3)
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x3, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x1, x5)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x3, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x5)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x4)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x4)

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

(43) Obligation:

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

SS_IN_GG(Xs, Ys) → U1_GG(Xs, Ys, perm_in_gg(Xs, Ys))
SS_IN_GG(Xs, Ys) → PERM_IN_GG(Xs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GG(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U6_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)
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_GG(Xs, X, Ys, perm_in_gg(Zs, Ys))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_GG(Xs, Ys, ordered_in_g(Ys))
U1_GG(Xs, Ys, perm_out_gg(Xs, Ys)) → ORDERED_IN_G(Ys)
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
ORDERED_IN_G(.(X, .(Y, Xs))) → LESS_IN_GG(X, s(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)
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg
SS_IN_GG(x1, x2)  =  SS_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x2, x3)
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x3, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x1, x5)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x3, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x5)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x4)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x3)
U8_G(x1, x2, x3, x4)  =  U8_G(x4)

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

(44) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 12 less nodes.

(45) Complex Obligation (AND)

(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)

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) Obligation:

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

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

(49) PiDPToQDPProof (EQUIVALENT transformation)

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

(50) Obligation:

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

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.

(51) 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

(52) TRUE

(53) Obligation:

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

U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg
ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)

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

(54) UsableRulesProof (EQUIVALENT transformation)

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

(55) Obligation:

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

U7_G(X, Y, Xs, less_out_gg(X, s(Y))) → ORDERED_IN_G(.(Y, Xs))
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(X, Y, Xs, less_in_gg(X, s(Y)))

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)
ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)

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

(56) PiDPToQDPProof (SOUND transformation)

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

(57) Obligation:

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

U7_G(Y, Xs, less_out_gg) → ORDERED_IN_G(.(Y, Xs))
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(Y, Xs, less_in_gg(X, s(Y)))

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.

(58) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

less_in_gg(0, s(X2)) → less_out_gg
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 2·x1 + 2·x2   
POL(0) = 2   
POL(ORDERED_IN_G(x1)) = 1 + x1   
POL(U7_G(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(U9_gg(x1)) = x1   
POL(less_in_gg(x1, x2)) = x1 + x2   
POL(less_out_gg) = 0   
POL(s(x1)) = x1   

(59) Obligation:

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

U7_G(Y, Xs, less_out_gg) → ORDERED_IN_G(.(Y, Xs))
ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(Y, Xs, less_in_gg(X, s(Y)))

The TRS R consists of the following rules:

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.

(60) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U7_G(Y, Xs, less_out_gg) → ORDERED_IN_G(.(Y, Xs))

Strictly oriented rules of the TRS R:

U9_gg(less_out_gg) → less_out_gg

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 2·x1 + 2·x2   
POL(ORDERED_IN_G(x1)) = 1 + x1   
POL(U7_G(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(U9_gg(x1)) = 2 + x1   
POL(less_in_gg(x1, x2)) = x1 + x2   
POL(less_out_gg) = 2   
POL(s(x1)) = 1 + x1   

(61) Obligation:

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

ORDERED_IN_G(.(X, .(Y, Xs))) → U7_G(Y, Xs, less_in_gg(X, s(Y)))

The TRS R consists of the following rules:

less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))

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.

(62) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(63) TRUE

(64) 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:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) 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

(67) PiDPToQDPProof (SOUND transformation)

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

(68) 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.

(69) 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

(70) TRUE

(71) Obligation:

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

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

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(72) UsableRulesProof (EQUIVALENT transformation)

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

(73) Obligation:

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

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

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

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

(74) PiDPToQDPProof (SOUND transformation)

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

(75) Obligation:

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

(76) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(77) TRUE

(78) Obligation:

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

U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

ss_in_gg(Xs, Ys) → U1_gg(Xs, Ys, perm_in_gg(Xs, Ys))
perm_in_gg([], []) → perm_out_gg([], [])
perm_in_gg(Xs, .(X, Ys)) → U3_gg(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_gg(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_gg(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_gg(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U5_gg(Xs, X, Ys, perm_in_gg(Zs, Ys))
U5_gg(Xs, X, Ys, perm_out_gg(Zs, Ys)) → perm_out_gg(Xs, .(X, Ys))
U1_gg(Xs, Ys, perm_out_gg(Xs, Ys)) → U2_gg(Xs, Ys, ordered_in_g(Ys))
ordered_in_g([]) → ordered_out_g([])
ordered_in_g(.(X1, [])) → ordered_out_g(.(X1, []))
ordered_in_g(.(X, .(Y, Xs))) → U7_g(X, Y, Xs, less_in_gg(X, s(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))
U7_g(X, Y, Xs, less_out_gg(X, s(Y))) → U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) → ordered_out_g(.(X, .(Y, Xs)))
U2_gg(Xs, Ys, ordered_out_g(Ys)) → ss_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_gg(x1, x2)  =  ss_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)
perm_in_gg(x1, x2)  =  perm_in_gg(x1, x2)
[]  =  []
perm_out_gg(x1, x2)  =  perm_out_gg
.(x1, x2)  =  .(x1, x2)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x3, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x3, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
U5_gg(x1, x2, x3, x4)  =  U5_gg(x4)
U2_gg(x1, x2, x3)  =  U2_gg(x3)
ordered_in_g(x1)  =  ordered_in_g(x1)
ordered_out_g(x1)  =  ordered_out_g
U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
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)
U8_g(x1, x2, x3, x4)  =  U8_g(x4)
ss_out_gg(x1, x2)  =  ss_out_gg
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x3, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x3, x4)

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

(79) UsableRulesProof (EQUIVALENT transformation)

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

(80) Obligation:

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

U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U6_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_aag([], X, X) → app_out_aag([], X, X)
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U6_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U6_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U6_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x1, x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x5)
PERM_IN_GG(x1, x2)  =  PERM_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x3, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x3, x4)

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

(81) PiDPToQDPProof (SOUND transformation)

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

(82) Obligation:

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

U3_GG(Ys, app_out_aag(X1s, .(X, X2s))) → U4_GG(Ys, app_in_gga(X1s, X2s))
U4_GG(Ys, app_out_gga(Zs)) → PERM_IN_GG(Zs, Ys)
PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Ys, app_in_aag(Xs))

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U6_gga(X, app_in_gga(Xs, Ys))
app_in_aag(X) → app_out_aag([], X)
app_in_aag(.(X, Zs)) → U6_aag(X, app_in_aag(Zs))
U6_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U6_aag(X, app_out_aag(Xs, Ys)) → app_out_aag(.(X, Xs), Ys)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U6_gga(x0, x1)
U6_aag(x0, x1)

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

(83) 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:

  • U4_GG(Ys, app_out_gga(Zs)) → PERM_IN_GG(Zs, Ys)
    The graph contains the following edges 2 > 1, 1 >= 2

  • PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Ys, app_in_aag(Xs))
    The graph contains the following edges 2 > 1

  • U3_GG(Ys, app_out_aag(X1s, .(X, X2s))) → U4_GG(Ys, app_in_gga(X1s, X2s))
    The graph contains the following edges 1 >= 1

(84) TRUE