Left Termination of the query pattern rev_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 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔)
        ↳18 QDP
  ↳19 PrologToPiTRSProof (⇐)
    ↳20 PiTRS
    ↳21 DependencyPairsProof (⇔)
    ↳22 PiDP
    ↳23 DependencyGraphProof (⇔)
    ↳24 AND
      ↳25 PiDP
        ↳26 UsableRulesProof (⇔)
        ↳27 PiDP
        ↳28 PiDPToQDPProof (⇐)
        ↳29 QDP
        ↳30 QDPSizeChangeProof (⇔)
        ↳31 TRUE
      ↳32 PiDP
        ↳33 PiDPToQDPProof (⇐)
        ↳34 QDP
        ↳35 QDPOrderProof (⇔)
        ↳36 QDP
        ↳37 DependencyGraphProof (⇔)
        ↳38 TRUE

(0) Obligation:

Clauses:

rev([], []).
rev(.(X, XS), .(Y, YS)) :- ','(rev1(X, XS, Y), rev2(X, XS, YS)).
rev1(X, [], X).
rev1(X, .(Y, YS), Z) :- rev1(Y, YS, Z).
rev2(X, [], []).
rev2(X, .(Y, YS), ZS) :- ','(rev2(Y, YS, US), ','(rev(US, VS), rev(.(X, VS), ZS))).

Queries:

rev(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:
rev_in: (b,f)
rev1_in: (b,b,f)
rev2_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)

(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:

REV_IN_GA(.(X, XS), .(Y, YS)) → U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
REV_IN_GA(.(X, XS), .(Y, YS)) → REV1_IN_GGA(X, XS, Y)
REV1_IN_GGA(X, .(Y, YS), Z) → U3_GGA(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_GA(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS, YS)
REV2_IN_GGA(X, .(Y, YS), ZS) → U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
REV2_IN_GGA(X, .(Y, YS), ZS) → REV2_IN_GGA(Y, YS, US)
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US, VS)
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_GGA(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS), ZS)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
REV1_IN_GGA(x1, x2, x3)  =  REV1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
REV2_IN_GGA(x1, x2, x3)  =  REV2_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x5)

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

(4) Obligation:

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

REV_IN_GA(.(X, XS), .(Y, YS)) → U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
REV_IN_GA(.(X, XS), .(Y, YS)) → REV1_IN_GGA(X, XS, Y)
REV1_IN_GGA(X, .(Y, YS), Z) → U3_GGA(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_GA(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS, YS)
REV2_IN_GGA(X, .(Y, YS), ZS) → U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
REV2_IN_GGA(X, .(Y, YS), ZS) → REV2_IN_GGA(Y, YS, US)
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US, VS)
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_GGA(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS), ZS)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
REV1_IN_GGA(x1, x2, x3)  =  REV1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
REV2_IN_GGA(x1, x2, x3)  =  REV2_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)
REV1_IN_GGA(x1, x2, x3)  =  REV1_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:

REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV1_IN_GGA(x1, x2, x3)  =  REV1_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:

REV1_IN_GGA(X, .(Y, YS)) → REV1_IN_GGA(Y, 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:

  • REV1_IN_GGA(X, .(Y, YS)) → REV1_IN_GGA(Y, YS)
    The graph contains the following edges 2 > 1, 2 > 2

(13) TRUE

(14) Obligation:

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

U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS, YS)
REV2_IN_GGA(X, .(Y, YS), ZS) → U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS), ZS)
REV_IN_GA(.(X, XS), .(Y, YS)) → U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US, VS)
REV2_IN_GGA(X, .(Y, YS), ZS) → REV2_IN_GGA(Y, YS, US)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x5)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
REV2_IN_GGA(x1, x2, x3)  =  REV2_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x5)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

U1_GA(X, XS, rev1_out_gga(Y)) → REV2_IN_GGA(X, XS)
REV2_IN_GGA(X, .(Y, YS)) → U4_GGA(X, rev2_in_gga(Y, YS))
U4_GGA(X, rev2_out_gga(US)) → U5_GGA(X, rev_in_ga(US))
U5_GGA(X, rev_out_ga(VS)) → REV_IN_GA(.(X, VS))
REV_IN_GA(.(X, XS)) → U1_GA(X, XS, rev1_in_gga(X, XS))
U4_GGA(X, rev2_out_gga(US)) → REV_IN_GA(US)
REV2_IN_GGA(X, .(Y, YS)) → REV2_IN_GGA(Y, YS)

The TRS R consists of the following rules:

rev_in_ga([]) → rev_out_ga([])
rev_in_ga(.(X, XS)) → U1_ga(X, XS, rev1_in_gga(X, XS))
rev1_in_gga(X, []) → rev1_out_gga(X)
rev1_in_gga(X, .(Y, YS)) → U3_gga(rev1_in_gga(Y, YS))
U3_gga(rev1_out_gga(Z)) → rev1_out_gga(Z)
U1_ga(X, XS, rev1_out_gga(Y)) → U2_ga(Y, rev2_in_gga(X, XS))
rev2_in_gga(X, []) → rev2_out_gga([])
rev2_in_gga(X, .(Y, YS)) → U4_gga(X, rev2_in_gga(Y, YS))
U4_gga(X, rev2_out_gga(US)) → U5_gga(X, rev_in_ga(US))
U5_gga(X, rev_out_ga(VS)) → U6_gga(rev_in_ga(.(X, VS)))
U6_gga(rev_out_ga(ZS)) → rev2_out_gga(ZS)
U2_ga(Y, rev2_out_gga(YS)) → rev_out_ga(.(Y, YS))

The set Q consists of the following terms:

rev_in_ga(x0)
rev1_in_gga(x0, x1)
U3_gga(x0)
U1_ga(x0, x1, x2)
rev2_in_gga(x0, x1)
U4_gga(x0, x1)
U5_gga(x0, x1)
U6_gga(x0)
U2_ga(x0, x1)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


REV_IN_GA(.(X, XS)) → U1_GA(X, XS, rev1_in_gga(X, XS))
U4_GGA(X, rev2_out_gga(US)) → REV_IN_GA(US)
REV2_IN_GGA(X, .(Y, YS)) → REV2_IN_GGA(Y, YS)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(REV2_IN_GGA(x1, x2)) = x2   
POL(REV_IN_GA(x1)) = x1   
POL(U1_GA(x1, x2, x3)) = x2   
POL(U1_ga(x1, x2, x3)) = 1 + x2   
POL(U2_ga(x1, x2)) = 1 + x2   
POL(U3_gga(x1)) = 0   
POL(U4_GGA(x1, x2)) = 1 + x2   
POL(U4_gga(x1, x2)) = 1 + x2   
POL(U5_GGA(x1, x2)) = 1 + x2   
POL(U5_gga(x1, x2)) = 1 + x2   
POL(U6_gga(x1)) = x1   
POL([]) = 0   
POL(rev1_in_gga(x1, x2)) = 0   
POL(rev1_out_gga(x1)) = 0   
POL(rev2_in_gga(x1, x2)) = x2   
POL(rev2_out_gga(x1)) = x1   
POL(rev_in_ga(x1)) = x1   
POL(rev_out_ga(x1)) = x1   

The following usable rules [FROCOS05] were oriented:

rev2_in_gga(X, []) → rev2_out_gga([])
rev2_in_gga(X, .(Y, YS)) → U4_gga(X, rev2_in_gga(Y, YS))
rev_in_ga([]) → rev_out_ga([])
rev_in_ga(.(X, XS)) → U1_ga(X, XS, rev1_in_gga(X, XS))
U1_ga(X, XS, rev1_out_gga(Y)) → U2_ga(Y, rev2_in_gga(X, XS))
U2_ga(Y, rev2_out_gga(YS)) → rev_out_ga(.(Y, YS))
U4_gga(X, rev2_out_gga(US)) → U5_gga(X, rev_in_ga(US))
U5_gga(X, rev_out_ga(VS)) → U6_gga(rev_in_ga(.(X, VS)))
U6_gga(rev_out_ga(ZS)) → rev2_out_gga(ZS)

(18) Obligation:

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

U1_GA(X, XS, rev1_out_gga(Y)) → REV2_IN_GGA(X, XS)
REV2_IN_GGA(X, .(Y, YS)) → U4_GGA(X, rev2_in_gga(Y, YS))
U4_GGA(X, rev2_out_gga(US)) → U5_GGA(X, rev_in_ga(US))
U5_GGA(X, rev_out_ga(VS)) → REV_IN_GA(.(X, VS))

The TRS R consists of the following rules:

rev_in_ga([]) → rev_out_ga([])
rev_in_ga(.(X, XS)) → U1_ga(X, XS, rev1_in_gga(X, XS))
rev1_in_gga(X, []) → rev1_out_gga(X)
rev1_in_gga(X, .(Y, YS)) → U3_gga(rev1_in_gga(Y, YS))
U3_gga(rev1_out_gga(Z)) → rev1_out_gga(Z)
U1_ga(X, XS, rev1_out_gga(Y)) → U2_ga(Y, rev2_in_gga(X, XS))
rev2_in_gga(X, []) → rev2_out_gga([])
rev2_in_gga(X, .(Y, YS)) → U4_gga(X, rev2_in_gga(Y, YS))
U4_gga(X, rev2_out_gga(US)) → U5_gga(X, rev_in_ga(US))
U5_gga(X, rev_out_ga(VS)) → U6_gga(rev_in_ga(.(X, VS)))
U6_gga(rev_out_ga(ZS)) → rev2_out_gga(ZS)
U2_ga(Y, rev2_out_gga(YS)) → rev_out_ga(.(Y, YS))

The set Q consists of the following terms:

rev_in_ga(x0)
rev1_in_gga(x0, x1)
U3_gga(x0)
U1_ga(x0, x1, x2)
rev2_in_gga(x0, x1)
U4_gga(x0, x1)
U5_gga(x0, x1)
U6_gga(x0)
U2_ga(x0, x1)

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

(19) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rev_in: (b,f)
rev1_in: (b,b,f)
rev2_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(20) Obligation:

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

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)

(21) 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:

REV_IN_GA(.(X, XS), .(Y, YS)) → U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
REV_IN_GA(.(X, XS), .(Y, YS)) → REV1_IN_GGA(X, XS, Y)
REV1_IN_GGA(X, .(Y, YS), Z) → U3_GGA(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_GA(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS, YS)
REV2_IN_GGA(X, .(Y, YS), ZS) → U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
REV2_IN_GGA(X, .(Y, YS), ZS) → REV2_IN_GGA(Y, YS, US)
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US, VS)
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_GGA(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS), ZS)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
REV1_IN_GGA(x1, x2, x3)  =  REV1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
REV2_IN_GGA(x1, x2, x3)  =  REV2_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)

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

(22) Obligation:

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

REV_IN_GA(.(X, XS), .(Y, YS)) → U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
REV_IN_GA(.(X, XS), .(Y, YS)) → REV1_IN_GGA(X, XS, Y)
REV1_IN_GGA(X, .(Y, YS), Z) → U3_GGA(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_GA(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS, YS)
REV2_IN_GGA(X, .(Y, YS), ZS) → U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
REV2_IN_GGA(X, .(Y, YS), ZS) → REV2_IN_GGA(Y, YS, US)
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US, VS)
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_GGA(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS), ZS)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
REV1_IN_GGA(x1, x2, x3)  =  REV1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
REV2_IN_GGA(x1, x2, x3)  =  REV2_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(24) Complex Obligation (AND)

(25) Obligation:

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

REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
REV1_IN_GGA(x1, x2, x3)  =  REV1_IN_GGA(x1, x2)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

REV1_IN_GGA(X, .(Y, YS), Z) → REV1_IN_GGA(Y, YS, Z)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV1_IN_GGA(x1, x2, x3)  =  REV1_IN_GGA(x1, x2)

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

REV1_IN_GGA(X, .(Y, YS)) → REV1_IN_GGA(Y, YS)

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

(30) 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:

  • REV1_IN_GGA(X, .(Y, YS)) → REV1_IN_GGA(Y, YS)
    The graph contains the following edges 2 > 1, 2 > 2

(31) TRUE

(32) Obligation:

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

U1_GA(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS, YS)
REV2_IN_GGA(X, .(Y, YS), ZS) → U4_GGA(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_GGA(X, Y, YS, ZS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS), ZS)
REV_IN_GA(.(X, XS), .(Y, YS)) → U1_GA(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
U4_GGA(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US, VS)
REV2_IN_GGA(X, .(Y, YS), ZS) → REV2_IN_GGA(Y, YS, US)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, XS), .(Y, YS)) → U1_ga(X, XS, Y, YS, rev1_in_gga(X, XS, Y))
rev1_in_gga(X, [], X) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS), Z) → U3_gga(X, Y, YS, Z, rev1_in_gga(Y, YS, Z))
U3_gga(X, Y, YS, Z, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, Y, YS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, YS, rev2_in_gga(X, XS, YS))
rev2_in_gga(X, [], []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS), ZS) → U4_gga(X, Y, YS, ZS, rev2_in_gga(Y, YS, US))
U4_gga(X, Y, YS, ZS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, ZS, rev_in_ga(US, VS))
U5_gga(X, Y, YS, ZS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, ZS, rev_in_ga(.(X, VS), ZS))
U6_gga(X, Y, YS, ZS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, YS, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
rev1_in_gga(x1, x2, x3)  =  rev1_in_gga(x1, x2)
rev1_out_gga(x1, x2, x3)  =  rev1_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
rev2_in_gga(x1, x2, x3)  =  rev2_in_gga(x1, x2)
rev2_out_gga(x1, x2, x3)  =  rev2_out_gga(x1, x2, x3)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
REV2_IN_GGA(x1, x2, x3)  =  REV2_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

U1_GA(X, XS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS)
REV2_IN_GGA(X, .(Y, YS)) → U4_GGA(X, Y, YS, rev2_in_gga(Y, YS))
U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, rev_in_ga(US))
U5_GGA(X, Y, YS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS))
REV_IN_GA(.(X, XS)) → U1_GA(X, XS, rev1_in_gga(X, XS))
U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US)
REV2_IN_GGA(X, .(Y, YS)) → REV2_IN_GGA(Y, YS)

The TRS R consists of the following rules:

rev_in_ga([]) → rev_out_ga([], [])
rev_in_ga(.(X, XS)) → U1_ga(X, XS, rev1_in_gga(X, XS))
rev1_in_gga(X, []) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS)) → U3_gga(X, Y, YS, rev1_in_gga(Y, YS))
U3_gga(X, Y, YS, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, rev2_in_gga(X, XS))
rev2_in_gga(X, []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS)) → U4_gga(X, Y, YS, rev2_in_gga(Y, YS))
U4_gga(X, Y, YS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, rev_in_ga(US))
U5_gga(X, Y, YS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, rev_in_ga(.(X, VS)))
U6_gga(X, Y, YS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The set Q consists of the following terms:

rev_in_ga(x0)
rev1_in_gga(x0, x1)
U3_gga(x0, x1, x2, x3)
U1_ga(x0, x1, x2)
rev2_in_gga(x0, x1)
U4_gga(x0, x1, x2, x3)
U5_gga(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3)
U2_ga(x0, x1, x2, x3)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


REV_IN_GA(.(X, XS)) → U1_GA(X, XS, rev1_in_gga(X, XS))
U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) → REV_IN_GA(US)
REV2_IN_GGA(X, .(Y, YS)) → REV2_IN_GGA(Y, YS)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(REV2_IN_GGA(x1, x2)) = x2   
POL(REV_IN_GA(x1)) = x1   
POL(U1_GA(x1, x2, x3)) = x2   
POL(U1_ga(x1, x2, x3)) = 1 + x2 + x3   
POL(U2_ga(x1, x2, x3, x4)) = 1 + x4   
POL(U3_gga(x1, x2, x3, x4)) = 1   
POL(U4_GGA(x1, x2, x3, x4)) = x4   
POL(U4_gga(x1, x2, x3, x4)) = 1 + x4   
POL(U5_GGA(x1, x2, x3, x4)) = x4   
POL(U5_gga(x1, x2, x3, x4)) = 1 + x4   
POL(U6_gga(x1, x2, x3, x4)) = x4   
POL([]) = 0   
POL(rev1_in_gga(x1, x2)) = 1   
POL(rev1_out_gga(x1, x2, x3)) = 1   
POL(rev2_in_gga(x1, x2)) = 1 + x2   
POL(rev2_out_gga(x1, x2, x3)) = 1 + x3   
POL(rev_in_ga(x1)) = 1 + x1   
POL(rev_out_ga(x1, x2)) = 1 + x2   

The following usable rules [FROCOS05] were oriented:

rev2_in_gga(X, []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS)) → U4_gga(X, Y, YS, rev2_in_gga(Y, YS))
rev_in_ga([]) → rev_out_ga([], [])
rev_in_ga(.(X, XS)) → U1_ga(X, XS, rev1_in_gga(X, XS))
rev1_in_gga(X, []) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS)) → U3_gga(X, Y, YS, rev1_in_gga(Y, YS))
U1_ga(X, XS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, rev2_in_gga(X, XS))
U2_ga(X, XS, Y, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))
U4_gga(X, Y, YS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, rev_in_ga(US))
U5_gga(X, Y, YS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, rev_in_ga(.(X, VS)))
U6_gga(X, Y, YS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U3_gga(X, Y, YS, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)

(36) Obligation:

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

U1_GA(X, XS, rev1_out_gga(X, XS, Y)) → REV2_IN_GGA(X, XS)
REV2_IN_GGA(X, .(Y, YS)) → U4_GGA(X, Y, YS, rev2_in_gga(Y, YS))
U4_GGA(X, Y, YS, rev2_out_gga(Y, YS, US)) → U5_GGA(X, Y, YS, rev_in_ga(US))
U5_GGA(X, Y, YS, rev_out_ga(US, VS)) → REV_IN_GA(.(X, VS))

The TRS R consists of the following rules:

rev_in_ga([]) → rev_out_ga([], [])
rev_in_ga(.(X, XS)) → U1_ga(X, XS, rev1_in_gga(X, XS))
rev1_in_gga(X, []) → rev1_out_gga(X, [], X)
rev1_in_gga(X, .(Y, YS)) → U3_gga(X, Y, YS, rev1_in_gga(Y, YS))
U3_gga(X, Y, YS, rev1_out_gga(Y, YS, Z)) → rev1_out_gga(X, .(Y, YS), Z)
U1_ga(X, XS, rev1_out_gga(X, XS, Y)) → U2_ga(X, XS, Y, rev2_in_gga(X, XS))
rev2_in_gga(X, []) → rev2_out_gga(X, [], [])
rev2_in_gga(X, .(Y, YS)) → U4_gga(X, Y, YS, rev2_in_gga(Y, YS))
U4_gga(X, Y, YS, rev2_out_gga(Y, YS, US)) → U5_gga(X, Y, YS, rev_in_ga(US))
U5_gga(X, Y, YS, rev_out_ga(US, VS)) → U6_gga(X, Y, YS, rev_in_ga(.(X, VS)))
U6_gga(X, Y, YS, rev_out_ga(.(X, VS), ZS)) → rev2_out_gga(X, .(Y, YS), ZS)
U2_ga(X, XS, Y, rev2_out_gga(X, XS, YS)) → rev_out_ga(.(X, XS), .(Y, YS))

The set Q consists of the following terms:

rev_in_ga(x0)
rev1_in_gga(x0, x1)
U3_gga(x0, x1, x2, x3)
U1_ga(x0, x1, x2)
rev2_in_gga(x0, x1)
U4_gga(x0, x1, x2, x3)
U5_gga(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3)
U2_ga(x0, x1, x2, x3)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(38) TRUE