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

0 Prolog
↳1 PrologToPiTRSProof (⇒, 61 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 69 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 19 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇔, 23 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇔, 0 ms)
        ↳18 QDP
        ↳19 MRRProof (⇔, 101 ms)
        ↳20 QDP
        ↳21 DependencyGraphProof (⇔, 0 ms)
        ↳22 TRUE
    ↳23 PiDP
        ↳24 UsableRulesProof (⇔, 0 ms)
        ↳25 PiDP
        ↳26 PiDPToQDPProof (⇒, 0 ms)
        ↳27 QDP
        ↳28 QDPSizeChangeProof (⇔, 0 ms)
        ↳29 YES
    ↳30 PiDP
        ↳31 UsableRulesProof (⇔, 0 ms)
        ↳32 PiDP
        ↳33 PiDPToQDPProof (⇒, 0 ms)
        ↳34 QDP
        ↳35 QDPSizeChangeProof (⇔, 0 ms)
        ↳36 YES
    ↳37 PiDP
        ↳38 UsableRulesProof (⇔, 0 ms)
        ↳39 PiDP
        ↳40 PiDPToQDPProof (⇒, 0 ms)
        ↳41 QDP
        ↳42 QDPSizeChangeProof (⇔, 0 ms)
        ↳43 YES

(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).

Query: ss(g,g)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. 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) YES

(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) YES

(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) YES

(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) 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:

U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s))
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(Xs))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U6_gga(x0, x1, x2, x3)
U6_aag(x0, x1, x2)

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:

  • U4_GG(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GG(Zs, Ys)
    The graph contains the following edges 4 > 1, 3 >= 2

  • PERM_IN_GG(Xs, .(X, Ys)) → U3_GG(Xs, X, Ys, app_in_aag(Xs))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

  • U3_GG(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U4_GG(Xs, X, Ys, app_in_gga(X1s, X2s))
    The graph contains the following edges 1 >= 1, 4 > 1, 2 >= 2, 4 > 2, 3 >= 3

(43) YES