Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

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

(0) Obligation:

Clauses:

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).

Queries:

perm(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

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

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)

(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:

PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x4, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)

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

(4) Obligation:

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

PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x4, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x2, x3, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

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

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(13) TRUE

(14) Obligation:

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

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

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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:

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)  =  .(x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(17) PiDPToQDPProof (SOUND 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:

APP_IN_AAG(.(Zs)) → APP_IN_AAG(Zs)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(20) TRUE

(21) Obligation:

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

U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x4, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(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)) → U4_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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_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:
[]  =  []
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2, x3)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(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)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x2, x3, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U1_GA(Xs, app_out_aag(X1s, .(X2s), Xs)) → U2_GA(Xs, app_in_gga(X1s, X2s))
U2_GA(Xs, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U1_GA(Xs, app_in_aag(Xs))

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga([], X, X)
app_in_gga(.(Xs), Ys) → U4_gga(Xs, Ys, app_in_gga(Xs, Ys))
app_in_aag(X) → app_out_aag([], X, X)
app_in_aag(.(Zs)) → U4_aag(Zs, app_in_aag(Zs))
U4_gga(Xs, Ys, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(Xs), Ys, .(Zs))
U4_aag(Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(Xs), Ys, .(Zs))

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U4_gga(x0, x1, x2)
U4_aag(x0, x1)

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

(26) PrologToPiTRSProof (SOUND transformation)

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

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(27) Obligation:

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

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)

(28) 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:

PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)

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

(29) Obligation:

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

PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP_IN_AAG(X1s, .(X, X2s), Xs)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → APP_IN_GGA(X1s, X2s, Zs)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U4_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)
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5)  =  U4_AAG(x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(31) Complex Obligation (AND)

(32) 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:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) 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)  =  .(x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) Obligation:

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

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

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

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

(38) TRUE

(39) 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:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) 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)  =  .(x2)
APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

APP_IN_AAG(.(Zs)) → APP_IN_AAG(Zs)

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

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

(45) TRUE

(46) Obligation:

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

U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, app_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(Xs, .(X, Ys)) → U1_ga(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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_ga(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)) → U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → U3_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
U3_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) Obligation:

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

U1_GA(Xs, X, Ys, app_out_aag(X1s, .(X, X2s), Xs)) → U2_GA(Xs, X, Ys, app_in_gga(X1s, X2s, Zs))
U2_GA(Xs, X, Ys, app_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U1_GA(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)) → U4_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)) → U4_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_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:
[]  =  []
app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
.(x1, x2)  =  .(x2)
app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
U4_aag(x1, x2, x3, x4, x5)  =  U4_aag(x5)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(49) PiDPToQDPProof (SOUND transformation)

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

(50) Obligation:

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

U1_GA(app_out_aag(X1s, .(X2s))) → U2_GA(app_in_gga(X1s, X2s))
U2_GA(app_out_gga(Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U1_GA(app_in_aag(Xs))

The TRS R consists of the following rules:

app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(Xs), Ys) → U4_gga(app_in_gga(Xs, Ys))
app_in_aag(X) → app_out_aag([], X)
app_in_aag(.(Zs)) → U4_aag(app_in_aag(Zs))
U4_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))
U4_aag(app_out_aag(Xs, Ys)) → app_out_aag(.(Xs), Ys)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U4_gga(x0)
U4_aag(x0)

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

(51) 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:

U1_GA(app_out_aag(X1s, .(X2s))) → U2_GA(app_in_gga(X1s, X2s))
U2_GA(app_out_gga(Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U1_GA(app_in_aag(Xs))

Strictly oriented rules of the TRS R:

app_in_gga([], X) → app_out_gga(X)
app_in_aag(X) → app_out_aag([], X)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 5 + x1   
POL(PERM_IN_GA(x1)) = 2 + x1   
POL(U1_GA(x1)) = x1   
POL(U2_GA(x1)) = x1   
POL(U4_aag(x1)) = 5 + x1   
POL(U4_gga(x1)) = 5 + x1   
POL([]) = 0   
POL(app_in_aag(x1)) = 1 + x1   
POL(app_in_gga(x1, x2)) = 4 + x1 + x2   
POL(app_out_aag(x1, x2)) = x1 + x2   
POL(app_out_gga(x1)) = 3 + x1   

(52) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app_in_gga(.(Xs), Ys) → U4_gga(app_in_gga(Xs, Ys))
app_in_aag(.(Zs)) → U4_aag(app_in_aag(Zs))
U4_gga(app_out_gga(Zs)) → app_out_gga(.(Zs))
U4_aag(app_out_aag(Xs, Ys)) → app_out_aag(.(Xs), Ys)

The set Q consists of the following terms:

app_in_gga(x0, x1)
app_in_aag(x0)
U4_gga(x0)
U4_aag(x0)

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

(53) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(54) TRUE