Left Termination of the query pattern ss_in_2(a, g) w.r.t. the given Prolog program could not be shown:

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 NonTerminationProof (⇔)
        ↳29 FALSE
      ↳30 PiDP
        ↳31 UsableRulesProof (⇔)
        ↳32 PiDP
        ↳33 PiDPToQDPProof (⇐)
        ↳34 QDP
        ↳35 QDPSizeChangeProof (⇔)
        ↳36 TRUE
  ↳37 PrologToPiTRSProof (⇐)
    ↳38 PiTRS
    ↳39 DependencyPairsProof (⇔)
    ↳40 PiDP
    ↳41 DependencyGraphProof (⇔)
    ↳42 AND
      ↳43 PiDP
        ↳44 UsableRulesProof (⇔)
        ↳45 PiDP
        ↳46 PiDPToQDPProof (⇔)
        ↳47 QDP
        ↳48 QDPSizeChangeProof (⇔)
        ↳49 TRUE
      ↳50 PiDP
        ↳51 UsableRulesProof (⇔)
        ↳52 PiDP
        ↳53 PiDPToQDPProof (⇐)
        ↳54 QDP
        ↳55 UsableRulesReductionPairsProof (⇔)
        ↳56 QDP
        ↳57 MRRProof (⇔)
        ↳58 QDP
        ↳59 DependencyGraphProof (⇔)
        ↳60 TRUE
      ↳61 PiDP
        ↳62 UsableRulesProof (⇔)
        ↳63 PiDP
        ↳64 PiDPToQDPProof (⇐)
        ↳65 QDP
        ↳66 NonTerminationProof (⇔)
        ↳67 FALSE
      ↳68 PiDP
        ↳69 UsableRulesProof (⇔)
        ↳70 PiDP
        ↳71 PiDPToQDPProof (⇐)
        ↳72 QDP
        ↳73 QDPSizeChangeProof (⇔)
        ↳74 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(a,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ss_in: (f,b)
perm_in: (f,b)
app_in: (f,f,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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(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_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(x2)
SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
U2_AG(x1, x2, x3)  =  U2_AG(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_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(x2)
SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
U2_AG(x1, x2, x3)  =  U2_AG(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 4 SCCs with 11 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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(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_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(x2)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

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

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

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

(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_AAA → APP_IN_AAA

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

(28) NonTerminationProof (EQUIVALENT transformation)

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

s = APP_IN_AAA evaluates to t =APP_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

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



(29) FALSE

(30) Obligation:

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

U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag(x2)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)

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:

U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)

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:

U3_AG(X, Ys, app_out_aaa) → U4_AG(X, Ys, app_in_aaa)
U4_AG(X, Ys, app_out_aaa) → PERM_IN_AG(Ys)
PERM_IN_AG(.(X, Ys)) → U3_AG(X, Ys, app_in_aaa)

The TRS R consists of the following rules:

app_in_aaa → app_out_aaa
app_in_aaa → U6_aaa(app_in_aaa)
U6_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U6_aaa(x0)

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:

  • U4_AG(X, Ys, app_out_aaa) → PERM_IN_AG(Ys)
    The graph contains the following edges 2 >= 1

  • PERM_IN_AG(.(X, Ys)) → U3_AG(X, Ys, app_in_aaa)
    The graph contains the following edges 1 > 1, 1 > 2

  • U3_AG(X, Ys, app_out_aaa) → U4_AG(X, Ys, app_in_aaa)
    The graph contains the following edges 1 >= 1, 2 >= 2

(36) TRUE

(37) 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: (f,b)
perm_in: (f,b)
app_in: (f,f,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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(38) Obligation:

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

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag

(39) 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_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag
SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
U2_AG(x1, x2, x3)  =  U2_AG(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

(40) Obligation:

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

SS_IN_AG(Xs, Ys) → U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
SS_IN_AG(Xs, Ys) → PERM_IN_AG(Xs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
PERM_IN_AG(Xs, .(X, Ys)) → APP_IN_AAA(X1s, .(X, X2s), Xs)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → APP_IN_AAA(X1s, X2s, Zs)
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_AG(Xs, Ys, ordered_in_g(Ys))
U1_AG(Xs, Ys, perm_out_ag(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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag
SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
U2_AG(x1, x2, x3)  =  U2_AG(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

(41) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 11 less nodes.

(42) Complex Obligation (AND)

(43) 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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) 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

(46) PiDPToQDPProof (EQUIVALENT transformation)

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

(47) 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.

(48) 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

(49) TRUE

(50) 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_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag
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

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) 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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) 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.

(55) 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   

(56) 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.

(57) 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   

(58) 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.

(59) DependencyGraphProof (EQUIVALENT transformation)

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

(60) TRUE

(61) Obligation:

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

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

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag
APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA

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

(62) UsableRulesProof (EQUIVALENT transformation)

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

(63) Obligation:

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

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

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

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

(64) PiDPToQDPProof (SOUND transformation)

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

(65) Obligation:

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

APP_IN_AAA → APP_IN_AAA

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

(66) NonTerminationProof (EQUIVALENT transformation)

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

s = APP_IN_AAA evaluates to t =APP_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

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



(67) FALSE

(68) Obligation:

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

U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

ss_in_ag(Xs, Ys) → U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(Xs, .(X, Ys)) → U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) → perm_out_ag(Xs, .(X, Ys))
U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) → U2_ag(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_ag(Xs, Ys, ordered_out_g(Ys)) → ss_out_ag(Xs, Ys)

The argument filtering Pi contains the following mapping:
ss_in_ag(x1, x2)  =  ss_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
U2_ag(x1, x2, x3)  =  U2_ag(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_ag(x1, x2)  =  ss_out_ag
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)

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

(69) UsableRulesProof (EQUIVALENT transformation)

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

(70) Obligation:

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

U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) → U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) → PERM_IN_AG(Zs, Ys)
PERM_IN_AG(Xs, .(X, Ys)) → U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app_in_aaa([], X, X) → app_out_aaa([], X, X)
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
app_in_aaa(x1, x2, x3)  =  app_in_aaa
app_out_aaa(x1, x2, x3)  =  app_out_aaa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)

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

(71) PiDPToQDPProof (SOUND transformation)

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

(72) Obligation:

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

U3_AG(Ys, app_out_aaa) → U4_AG(Ys, app_in_aaa)
U4_AG(Ys, app_out_aaa) → PERM_IN_AG(Ys)
PERM_IN_AG(.(X, Ys)) → U3_AG(Ys, app_in_aaa)

The TRS R consists of the following rules:

app_in_aaa → app_out_aaa
app_in_aaa → U6_aaa(app_in_aaa)
U6_aaa(app_out_aaa) → app_out_aaa

The set Q consists of the following terms:

app_in_aaa
U6_aaa(x0)

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

(73) 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_AG(Ys, app_out_aaa) → PERM_IN_AG(Ys)
    The graph contains the following edges 1 >= 1

  • PERM_IN_AG(.(X, Ys)) → U3_AG(Ys, app_in_aaa)
    The graph contains the following edges 1 > 1

  • U3_AG(Ys, app_out_aaa) → U4_AG(Ys, app_in_aaa)
    The graph contains the following edges 1 >= 1

(74) TRUE