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:

app1(.(X0, X), Y, .(X0, Z)) :- app1(X, Y, Z).
app1([], Y, Y).
app2(.(X0, X), Y, .(X0, Z)) :- app2(X, Y, Z).
app2([], Y, Y).
perm(X, .(X0, Y)) :- ','(app1(X1, .(X0, X2), X), ','(app2(X1, X2, Z), perm(Z, Y))).
perm([], []).

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)
app1_in: (f,f,b)
app2_in: (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(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)

(3) DependencyPairsProof (EQUIVALENT transformation)

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

PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
PERM_IN_GA(X, .(X0, Y)) → APP1_IN_AAG(X1, .(X0, X2), X)
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z))
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → APP2_IN_GGA(X1, X2, Z)
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z))
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_GA(X, X0, Y, perm_in_ga(Z, Y))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
APP1_IN_AAG(x1, x2, x3)  =  APP1_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x4, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
APP2_IN_GGA(x1, x2, x3)  =  APP2_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x2, x3, x5)
U5_GA(x1, x2, x3, x4)  =  U5_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(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
PERM_IN_GA(X, .(X0, Y)) → APP1_IN_AAG(X1, .(X0, X2), X)
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z))
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → APP2_IN_GGA(X1, X2, Z)
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z))
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_GA(X, X0, Y, perm_in_ga(Z, Y))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
APP1_IN_AAG(x1, x2, x3)  =  APP1_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x4, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
APP2_IN_GGA(x1, x2, x3)  =  APP2_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x2, x3, x5)
U5_GA(x1, x2, x3, x4)  =  U5_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:

APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
APP2_IN_GGA(x1, x2, x3)  =  APP2_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:

APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP2_IN_GGA(x1, x2, x3)  =  APP2_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:

APP2_IN_GGA(.(X), Y) → APP2_IN_GGA(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:

  • APP2_IN_GGA(.(X), Y) → APP2_IN_GGA(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:

APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
APP1_IN_AAG(x1, x2, x3)  =  APP1_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:

APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP1_IN_AAG(x1, x2, x3)  =  APP1_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:

APP1_IN_AAG(.(Z)) → APP1_IN_AAG(Z)

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:

  • APP1_IN_AAG(.(Z)) → APP1_IN_AAG(Z)
    The graph contains the following edges 1 > 1

(20) TRUE

(21) Obligation:

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

U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)
PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)
PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))

The TRS R consists of the following rules:

app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))

The argument filtering Pi contains the following mapping:
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x4, x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2, x3)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x3, x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x1, x2, x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U3_GA(X, app1_out_aag(X1, .(X2), X)) → U4_GA(X, app2_in_gga(X1, X2))
U4_GA(X, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z)
PERM_IN_GA(X) → U3_GA(X, app1_in_aag(X))

The TRS R consists of the following rules:

app2_in_gga(.(X), Y) → U2_gga(X, Y, app2_in_gga(X, Y))
app2_in_gga([], Y) → app2_out_gga([], Y, Y)
app1_in_aag(.(Z)) → U1_aag(Z, app1_in_aag(Z))
app1_in_aag(Y) → app1_out_aag([], Y, Y)
U2_gga(X, Y, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X), Y, .(Z))
U1_aag(Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X), Y, .(Z))

The set Q consists of the following terms:

app2_in_gga(x0, x1)
app1_in_aag(x0)
U2_gga(x0, x1, x2)
U1_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)
app1_in: (f,f,b)
app2_in: (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(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)

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(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)

(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(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
PERM_IN_GA(X, .(X0, Y)) → APP1_IN_AAG(X1, .(X0, X2), X)
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z))
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → APP2_IN_GGA(X1, X2, Z)
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z))
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_GA(X, X0, Y, perm_in_ga(Z, Y))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
APP1_IN_AAG(x1, x2, x3)  =  APP1_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
APP2_IN_GGA(x1, x2, x3)  =  APP2_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
U5_GA(x1, x2, x3, x4)  =  U5_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(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
PERM_IN_GA(X, .(X0, Y)) → APP1_IN_AAG(X1, .(X0, X2), X)
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → U1_AAG(X0, X, Y, Z, app1_in_aag(X, Y, Z))
APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → APP2_IN_GGA(X1, X2, Z)
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → U2_GGA(X0, X, Y, Z, app2_in_gga(X, Y, Z))
APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_GA(X, X0, Y, perm_in_ga(Z, Y))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
APP1_IN_AAG(x1, x2, x3)  =  APP1_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
APP2_IN_GGA(x1, x2, x3)  =  APP2_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
U5_GA(x1, x2, x3, x4)  =  U5_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:

APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
APP2_IN_GGA(x1, x2, x3)  =  APP2_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:

APP2_IN_GGA(.(X0, X), Y, .(X0, Z)) → APP2_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP2_IN_GGA(x1, x2, x3)  =  APP2_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:

APP2_IN_GGA(.(X), Y) → APP2_IN_GGA(X, Y)

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:

  • APP2_IN_GGA(.(X), Y) → APP2_IN_GGA(X, Y)
    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:

APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
APP1_IN_AAG(x1, x2, x3)  =  APP1_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:

APP1_IN_AAG(.(X0, X), Y, .(X0, Z)) → APP1_IN_AAG(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP1_IN_AAG(x1, x2, x3)  =  APP1_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:

APP1_IN_AAG(.(Z)) → APP1_IN_AAG(Z)

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:

  • APP1_IN_AAG(.(Z)) → APP1_IN_AAG(Z)
    The graph contains the following edges 1 > 1

(45) TRUE

(46) Obligation:

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

U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)
PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))

The TRS R consists of the following rules:

perm_in_ga(X, .(X0, Y)) → U3_ga(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))
U3_ga(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_ga(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U4_ga(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → U5_ga(X, X0, Y, perm_in_ga(Z, Y))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(X, X0, Y, perm_out_ga(Z, Y)) → perm_out_ga(X, .(X0, Y))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)

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:

U3_GA(X, X0, Y, app1_out_aag(X1, .(X0, X2), X)) → U4_GA(X, X0, Y, X1, X2, app2_in_gga(X1, X2, Z))
U4_GA(X, X0, Y, X1, X2, app2_out_gga(X1, X2, Z)) → PERM_IN_GA(Z, Y)
PERM_IN_GA(X, .(X0, Y)) → U3_GA(X, X0, Y, app1_in_aag(X1, .(X0, X2), X))

The TRS R consists of the following rules:

app2_in_gga(.(X0, X), Y, .(X0, Z)) → U2_gga(X0, X, Y, Z, app2_in_gga(X, Y, Z))
app2_in_gga([], Y, Y) → app2_out_gga([], Y, Y)
app1_in_aag(.(X0, X), Y, .(X0, Z)) → U1_aag(X0, X, Y, Z, app1_in_aag(X, Y, Z))
app1_in_aag([], Y, Y) → app1_out_aag([], Y, Y)
U2_gga(X0, X, Y, Z, app2_out_gga(X, Y, Z)) → app2_out_gga(.(X0, X), Y, .(X0, Z))
U1_aag(X0, X, Y, Z, app1_out_aag(X, Y, Z)) → app1_out_aag(.(X0, X), Y, .(X0, Z))

The argument filtering Pi contains the following mapping:
app1_in_aag(x1, x2, x3)  =  app1_in_aag(x3)
.(x1, x2)  =  .(x2)
U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
app1_out_aag(x1, x2, x3)  =  app1_out_aag(x1, x2)
app2_in_gga(x1, x2, x3)  =  app2_in_gga(x1, x2)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
app2_out_gga(x1, x2, x3)  =  app2_out_gga(x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)

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:

U3_GA(app1_out_aag(X1, .(X2))) → U4_GA(app2_in_gga(X1, X2))
U4_GA(app2_out_gga(Z)) → PERM_IN_GA(Z)
PERM_IN_GA(X) → U3_GA(app1_in_aag(X))

The TRS R consists of the following rules:

app2_in_gga(.(X), Y) → U2_gga(app2_in_gga(X, Y))
app2_in_gga([], Y) → app2_out_gga(Y)
app1_in_aag(.(Z)) → U1_aag(app1_in_aag(Z))
app1_in_aag(Y) → app1_out_aag([], Y)
U2_gga(app2_out_gga(Z)) → app2_out_gga(.(Z))
U1_aag(app1_out_aag(X, Y)) → app1_out_aag(.(X), Y)

The set Q consists of the following terms:

app2_in_gga(x0, x1)
app1_in_aag(x0)
U2_gga(x0)
U1_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:

U3_GA(app1_out_aag(X1, .(X2))) → U4_GA(app2_in_gga(X1, X2))
U4_GA(app2_out_gga(Z)) → PERM_IN_GA(Z)
PERM_IN_GA(X) → U3_GA(app1_in_aag(X))

Strictly oriented rules of the TRS R:

app2_in_gga([], Y) → app2_out_gga(Y)
app1_in_aag(Y) → app1_out_aag([], Y)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 5 + x1   
POL(PERM_IN_GA(x1)) = 2 + x1   
POL(U1_aag(x1)) = 5 + x1   
POL(U2_gga(x1)) = 5 + x1   
POL(U3_GA(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL([]) = 0   
POL(app1_in_aag(x1)) = 1 + x1   
POL(app1_out_aag(x1, x2)) = x1 + x2   
POL(app2_in_gga(x1, x2)) = 4 + x1 + x2   
POL(app2_out_gga(x1)) = 3 + x1   

(52) Obligation:

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

app2_in_gga(.(X), Y) → U2_gga(app2_in_gga(X, Y))
app1_in_aag(.(Z)) → U1_aag(app1_in_aag(Z))
U2_gga(app2_out_gga(Z)) → app2_out_gga(.(Z))
U1_aag(app1_out_aag(X, Y)) → app1_out_aag(.(X), Y)

The set Q consists of the following terms:

app2_in_gga(x0, x1)
app1_in_aag(x0)
U2_gga(x0)
U1_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