Left Termination proof of /tmp/tmp6sZDns/rotate.pl

Left Termination of the query pattern rotate_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 PrologToPiTRSProof (⇐)

    ↳22 PiTRS

    ↳23 DependencyPairsProof (⇔)

    ↳24 PiDP

    ↳25 DependencyGraphProof (⇔)

    ↳26 AND

      ↳27 PiDP

        ↳28 UsableRulesProof (⇔)

        ↳29 PiDP

        ↳30 PiDPToQDPProof (⇐)

        ↳31 QDP

        ↳32 QDPSizeChangeProof (⇔)

        ↳33 TRUE

      ↳34 PiDP

        ↳35 UsableRulesProof (⇔)

        ↳36 PiDP

        ↳37 PiDPToQDPProof (⇐)

        ↳38 QDP

(0) Obligation:

Clauses:

rotate(X, Y) :- ','(append2(A, B, X), append1(B, A, Y)).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append1([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
append2([], Ys, Ys).

Queries:

rotate(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:
rotate_in: (b,f)
append2_in: (f,f,b)
append1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

(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:

ROTATE_IN_GA(X, Y) → U1_GA(X, Y, append2_in_aag(A, B, X))
ROTATE_IN_GA(X, Y) → APPEND2_IN_AAG(A, B, X)
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)
U1_GA(X, Y, append2_out_aag(A, B, X)) → U2_GA(X, Y, append1_in_gga(B, A, Y))
U1_GA(X, Y, append2_out_aag(A, B, X)) → APPEND1_IN_GGA(B, A, Y)
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)
ROTATE_IN_GA(x1, x2) = ROTATE_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5) = U4_AAG(x1, x5)
U2_GA(x1, x2, x3) = U2_GA(x3)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x5)

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

(4) Obligation:

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

ROTATE_IN_GA(X, Y) → U1_GA(X, Y, append2_in_aag(A, B, X))
ROTATE_IN_GA(X, Y) → APPEND2_IN_AAG(A, B, X)
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)
U1_GA(X, Y, append2_out_aag(A, B, X)) → U2_GA(X, Y, append1_in_gga(B, A, Y))
U1_GA(X, Y, append2_out_aag(A, B, X)) → APPEND1_IN_GGA(B, A, Y)
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_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:

APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_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:

APPEND1_IN_GGA(.(X, Xs), Ys) → APPEND1_IN_GGA(Xs, Ys)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

APPEND1_IN_GGA(.(X, Xs), Ys) → APPEND1_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

(13) TRUE

(14) Obligation:

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

APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_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:

APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_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:

APPEND2_IN_AAG(.(X, Zs)) → APPEND2_IN_AAG(Zs)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

APPEND2_IN_AAG(.(X, Zs)) → APPEND2_IN_AAG(Zs)
The graph contains the following edges 1 > 1

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

(23) 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:

ROTATE_IN_GA(X, Y) → U1_GA(X, Y, append2_in_aag(A, B, X))
ROTATE_IN_GA(X, Y) → APPEND2_IN_AAG(A, B, X)
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)
U1_GA(X, Y, append2_out_aag(A, B, X)) → U2_GA(X, Y, append1_in_gga(B, A, Y))
U1_GA(X, Y, append2_out_aag(A, B, X)) → APPEND1_IN_GGA(B, A, Y)
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x4, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2, x3)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x1, x2, x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x1, x2)
ROTATE_IN_GA(x1, x2) = ROTATE_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5) = U4_AAG(x1, x4, x5)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5)

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

(24) Obligation:

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

ROTATE_IN_GA(X, Y) → U1_GA(X, Y, append2_in_aag(A, B, X))
ROTATE_IN_GA(X, Y) → APPEND2_IN_AAG(A, B, X)
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)
U1_GA(X, Y, append2_out_aag(A, B, X)) → U2_GA(X, Y, append1_in_gga(B, A, Y))
U1_GA(X, Y, append2_out_aag(A, B, X)) → APPEND1_IN_GGA(B, A, Y)
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x4, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2, x3)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x1, x2, x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x1, x2)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2)

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

APPEND1_IN_GGA(.(X, Xs), Ys) → APPEND1_IN_GGA(Xs, Ys)

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

(32) 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:

APPEND1_IN_GGA(.(X, Xs), Ys) → APPEND1_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

(33) TRUE

(34) Obligation:

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

APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x4, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2, x3)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x1, x2, x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x1, x2)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

APPEND2_IN_AAG(.(X, Zs)) → APPEND2_IN_AAG(Zs)

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