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:

append(nil, XS, XS).
append(cons(X, XS1), XS2, cons(X, YS)) :- append(XS1, XS2, YS).
split(XS, nil, XS).
split(cons(X, XS), cons(X, YS1), YS2) :- split(XS, YS1, YS2).
perm(nil, nil).
perm(XS, cons(Y, YS)) :- ','(split(XS, YS1, cons(Y, YS2)), ','(append(YS1, YS2, ZS), perm(ZS, YS))).

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)
split_in: (b,f,f)
append_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(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_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(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_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, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAA(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → U2_GAA(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → APPEND_IN_GGA(YS1, YS2, ZS)
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → U1_GGA(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_ga(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_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(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAA(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → U2_GAA(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → APPEND_IN_GGA(YS1, YS2, ZS)
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → U1_GGA(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_ga(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x2, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_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:

APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_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:

APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)

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

APPEND_IN_GGA(cons(XS1), XS2) → APPEND_IN_GGA(XS1, XS2)

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:

  • APPEND_IN_GGA(cons(XS1), XS2) → APPEND_IN_GGA(XS1, XS2)
    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:

SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)

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:

SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)

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

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:

SPLIT_IN_GAA(cons(XS)) → SPLIT_IN_GAA(XS)

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:

  • SPLIT_IN_GAA(cons(XS)) → SPLIT_IN_GAA(XS)
    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(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)
PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
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(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)
PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)

The argument filtering Pi contains the following mapping:
nil  =  nil
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x1, x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x2, x5)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
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(XS, split_out_gaa(XS, YS1, cons(YS2))) → U4_GA(XS, append_in_gga(YS1, YS2))
U4_GA(XS, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS)
PERM_IN_GA(XS) → U3_GA(XS, split_in_gaa(XS))

The TRS R consists of the following rules:

append_in_gga(nil, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(XS1), XS2) → U1_gga(XS1, XS2, append_in_gga(XS1, XS2))
split_in_gaa(XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(XS)) → U2_gaa(XS, split_in_gaa(XS))
U1_gga(XS1, XS2, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(XS1), XS2, cons(YS))
U2_gaa(XS, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(XS), cons(YS1), YS2)

The set Q consists of the following terms:

append_in_gga(x0, x1)
split_in_gaa(x0)
U1_gga(x0, x1, x2)
U2_gaa(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)
split_in: (b,f,f)
append_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(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
U5_ga(x1, x2, x3, x4)  =  U5_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(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
U5_ga(x1, x2, x3, x4)  =  U5_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, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAA(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → U2_GAA(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → APPEND_IN_GGA(YS1, YS2, ZS)
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → U1_GGA(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_ga(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_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(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
PERM_IN_GA(XS, cons(Y, YS)) → SPLIT_IN_GAA(XS, YS1, cons(Y, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → U2_GAA(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U3_GA(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → APPEND_IN_GGA(YS1, YS2, ZS)
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → U1_GGA(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_GA(XS, Y, YS, perm_in_ga(ZS, YS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5)  =  U2_GAA(x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_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:

APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
APPEND_IN_GGA(x1, x2, x3)  =  APPEND_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:

APPEND_IN_GGA(cons(X, XS1), XS2, cons(X, YS)) → APPEND_IN_GGA(XS1, XS2, YS)

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

APPEND_IN_GGA(cons(XS1), XS2) → APPEND_IN_GGA(XS1, XS2)

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:

  • APPEND_IN_GGA(cons(XS1), XS2) → APPEND_IN_GGA(XS1, XS2)
    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:

SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
SPLIT_IN_GAA(x1, x2, x3)  =  SPLIT_IN_GAA(x1)

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:

SPLIT_IN_GAA(cons(X, XS), cons(X, YS1), YS2) → SPLIT_IN_GAA(XS, YS1, YS2)

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

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:

SPLIT_IN_GAA(cons(XS)) → SPLIT_IN_GAA(XS)

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:

  • SPLIT_IN_GAA(cons(XS)) → SPLIT_IN_GAA(XS)
    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(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)
PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

perm_in_ga(nil, nil) → perm_out_ga(nil, nil)
perm_in_ga(XS, cons(Y, YS)) → U3_ga(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)
U3_ga(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_ga(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U4_ga(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → U5_ga(XS, Y, YS, perm_in_ga(ZS, YS))
U5_ga(XS, Y, YS, perm_out_ga(ZS, YS)) → perm_out_ga(XS, cons(Y, YS))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
nil  =  nil
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
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(XS, Y, YS, split_out_gaa(XS, YS1, cons(Y, YS2))) → U4_GA(XS, Y, YS, YS1, YS2, append_in_gga(YS1, YS2, ZS))
U4_GA(XS, Y, YS, YS1, YS2, append_out_gga(YS1, YS2, ZS)) → PERM_IN_GA(ZS, YS)
PERM_IN_GA(XS, cons(Y, YS)) → U3_GA(XS, Y, YS, split_in_gaa(XS, YS1, cons(Y, YS2)))

The TRS R consists of the following rules:

append_in_gga(nil, XS, XS) → append_out_gga(nil, XS, XS)
append_in_gga(cons(X, XS1), XS2, cons(X, YS)) → U1_gga(X, XS1, XS2, YS, append_in_gga(XS1, XS2, YS))
split_in_gaa(XS, nil, XS) → split_out_gaa(XS, nil, XS)
split_in_gaa(cons(X, XS), cons(X, YS1), YS2) → U2_gaa(X, XS, YS1, YS2, split_in_gaa(XS, YS1, YS2))
U1_gga(X, XS1, XS2, YS, append_out_gga(XS1, XS2, YS)) → append_out_gga(cons(X, XS1), XS2, cons(X, YS))
U2_gaa(X, XS, YS1, YS2, split_out_gaa(XS, YS1, YS2)) → split_out_gaa(cons(X, XS), cons(X, YS1), YS2)

The argument filtering Pi contains the following mapping:
nil  =  nil
split_in_gaa(x1, x2, x3)  =  split_in_gaa(x1)
cons(x1, x2)  =  cons(x2)
split_out_gaa(x1, x2, x3)  =  split_out_gaa(x2, x3)
U2_gaa(x1, x2, x3, x4, x5)  =  U2_gaa(x5)
append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
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(split_out_gaa(YS1, cons(YS2))) → U4_GA(append_in_gga(YS1, YS2))
U4_GA(append_out_gga(ZS)) → PERM_IN_GA(ZS)
PERM_IN_GA(XS) → U3_GA(split_in_gaa(XS))

The TRS R consists of the following rules:

append_in_gga(nil, XS) → append_out_gga(XS)
append_in_gga(cons(XS1), XS2) → U1_gga(append_in_gga(XS1, XS2))
split_in_gaa(XS) → split_out_gaa(nil, XS)
split_in_gaa(cons(XS)) → U2_gaa(split_in_gaa(XS))
U1_gga(append_out_gga(YS)) → append_out_gga(cons(YS))
U2_gaa(split_out_gaa(YS1, YS2)) → split_out_gaa(cons(YS1), YS2)

The set Q consists of the following terms:

append_in_gga(x0, x1)
split_in_gaa(x0)
U1_gga(x0)
U2_gaa(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(split_out_gaa(YS1, cons(YS2))) → U4_GA(append_in_gga(YS1, YS2))
U4_GA(append_out_gga(ZS)) → PERM_IN_GA(ZS)
PERM_IN_GA(XS) → U3_GA(split_in_gaa(XS))

Strictly oriented rules of the TRS R:

append_in_gga(nil, XS) → append_out_gga(XS)
split_in_gaa(XS) → split_out_gaa(nil, XS)

Used ordering: Polynomial interpretation [POLO]:

POL(PERM_IN_GA(x1)) = 2 + x1   
POL(U1_gga(x1)) = 5 + x1   
POL(U2_gaa(x1)) = 5 + x1   
POL(U3_GA(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL(append_in_gga(x1, x2)) = 4 + x1 + x2   
POL(append_out_gga(x1)) = 3 + x1   
POL(cons(x1)) = 5 + x1   
POL(nil) = 0   
POL(split_in_gaa(x1)) = 1 + x1   
POL(split_out_gaa(x1, x2)) = x1 + x2   

(52) Obligation:

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

append_in_gga(cons(XS1), XS2) → U1_gga(append_in_gga(XS1, XS2))
split_in_gaa(cons(XS)) → U2_gaa(split_in_gaa(XS))
U1_gga(append_out_gga(YS)) → append_out_gga(cons(YS))
U2_gaa(split_out_gaa(YS1, YS2)) → split_out_gaa(cons(YS1), YS2)

The set Q consists of the following terms:

append_in_gga(x0, x1)
split_in_gaa(x0)
U1_gga(x0)
U2_gaa(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