Left Termination proof of /tmp/tmpZEpgob/rev.pl

Left Termination of the query pattern rev_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 CutEliminatorProof (⇐)

↳2 Prolog

  ↳3 PrologToPiTRSProof (⇐)

    ↳4 PiTRS

    ↳5 DependencyPairsProof (⇔)

    ↳6 PiDP

    ↳7 DependencyGraphProof (⇔)

    ↳8 PiDP

    ↳9 UsableRulesProof (⇔)

    ↳10 PiDP

    ↳11 PiDPToQDPProof (⇐)

    ↳12 QDP

      ↳13 Instantiation (⇔)

        ↳14 QDP

      ↳15 Narrowing (⇐)

        ↳16 QDP

        ↳17 UsableRulesProof (⇔)

        ↳18 QDP

        ↳19 QReductionProof (⇔)

        ↳20 QDP

        ↳21 Narrowing (⇐)

        ↳22 QDP

        ↳23 UsableRulesProof (⇔)

        ↳24 QDP

        ↳25 QReductionProof (⇔)

        ↳26 QDP

        ↳27 QDPOrderProof (⇔)

        ↳28 QDP

        ↳29 DependencyGraphProof (⇔)

        ↳30 QDP

          ↳31 Instantiation (⇔)

            ↳32 QDP

          ↳33 Instantiation (⇔)

            ↳34 QDP

              ↳35 Instantiation (⇔)

                ↳36 QDP

              ↳37 Instantiation (⇔)

                ↳38 QDP

                ↳39 Instantiation (⇔)

                ↳40 QDP

                ↳41 Instantiation (⇔)

                ↳42 QDP

                ↳43 Instantiation (⇔)

                ↳44 QDP

                ↳45 Instantiation (⇔)

                ↳46 QDP

                ↳47 NonTerminationProof (⇔)

                ↳48 FALSE

  ↳49 PrologToPiTRSProof (⇐)

    ↳50 PiTRS

    ↳51 DependencyPairsProof (⇔)

    ↳52 PiDP

    ↳53 DependencyGraphProof (⇔)

    ↳54 PiDP

    ↳55 UsableRulesProof (⇔)

    ↳56 PiDP

    ↳57 PiDPToQDPProof (⇐)

    ↳58 QDP

      ↳59 Instantiation (⇔)

        ↳60 QDP

      ↳61 Narrowing (⇐)

        ↳62 QDP

        ↳63 UsableRulesProof (⇔)

        ↳64 QDP

        ↳65 QReductionProof (⇔)

        ↳66 QDP

        ↳67 Narrowing (⇐)

        ↳68 QDP

        ↳69 UsableRulesProof (⇔)

        ↳70 QDP

        ↳71 QReductionProof (⇔)

        ↳72 QDP

        ↳73 Instantiation (⇔)

        ↳74 QDP

        ↳75 DependencyGraphProof (⇔)

        ↳76 AND

          ↳77 QDP

            ↳78 Instantiation (⇔)

              ↳79 QDP

            ↳80 Instantiation (⇔)

              ↳81 QDP

              ↳82 Instantiation (⇔)

              ↳83 QDP

              ↳84 Instantiation (⇔)

              ↳85 QDP

              ↳86 Instantiation (⇔)

              ↳87 QDP

              ↳88 Instantiation (⇔)

              ↳89 QDP

              ↳90 NonTerminationProof (⇔)

              ↳91 FALSE

          ↳92 QDP

            ↳93 QDPSizeChangeProof (⇔)

            ↳94 TRUE

(0) Obligation:

Clauses:

rev(L, R) :- rev(L, [], R).
rev([], Y, Z) :- ','(!, eq(Y, Z)).
rev(L, S, R) :- ','(head(L, X), ','(tail(L, T), rev(T, .(X, S), R))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
eq(X, X).

Queries:

rev(g,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

rev(L, R) :- rev(L, [], R).
rev([], Y, Z) :- eq(Y, Z).
rev(L, S, R) :- ','(head(L, X), ','(tail(L, T), rev(T, .(X, S), R))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
eq(X, X).

Queries:

rev(g,a).

(3) 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)
rev_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(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

(5) 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(L, R) → U1_GA(L, R, rev_in_gga(L, [], R))
REV_IN_GA(L, R) → REV_IN_GGA(L, [], R)
REV_IN_GGA([], Y, Z) → U2_GGA(Y, Z, eq_in_ga(Y, Z))
REV_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
REV_IN_GGA(L, S, R) → HEAD_IN_GA(L, X)
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U3_GGA(L, S, R, head_out_ga(L, X)) → TAIL_IN_GA(L, T)
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → U5_GGA(L, S, R, rev_in_gga(T, .(X, S), R))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2) = rev_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2)
[] = []
U2_gga(x1, x2, x3) = U2_gga(x3)
eq_in_ga(x1, x2) = eq_in_ga(x1)
eq_out_ga(x1, x2) = eq_out_ga(x2)
rev_out_gga(x1, x2, x3) = rev_out_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga
.(x1, x2) = .(x2)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
rev_out_ga(x1, x2) = rev_out_ga(x2)
REV_IN_GA(x1, x2) = REV_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3) = U2_GGA(x3)
EQ_IN_GA(x1, x2) = EQ_IN_GA(x1)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
HEAD_IN_GA(x1, x2) = HEAD_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x2, x5)
TAIL_IN_GA(x1, x2) = TAIL_IN_GA(x1)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x4)

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

(6) Obligation:

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

REV_IN_GA(L, R) → U1_GA(L, R, rev_in_gga(L, [], R))
REV_IN_GA(L, R) → REV_IN_GGA(L, [], R)
REV_IN_GGA([], Y, Z) → U2_GGA(Y, Z, eq_in_ga(Y, Z))
REV_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
REV_IN_GGA(L, S, R) → HEAD_IN_GA(L, X)
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U3_GGA(L, S, R, head_out_ga(L, X)) → TAIL_IN_GA(L, T)
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → U5_GGA(L, S, R, rev_in_gga(T, .(X, S), R))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.

(8) Obligation:

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

REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2) = rev_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2)
[] = []
U2_gga(x1, x2, x3) = U2_gga(x3)
eq_in_ga(x1, x2) = eq_in_ga(x1)
eq_out_ga(x1, x2) = eq_out_ga(x2)
rev_out_gga(x1, x2, x3) = rev_out_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga
.(x1, x2) = .(x2)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x2, x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
rev_out_ga(x1, x2) = rev_out_ga(x2)
REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x2, x5)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)

The argument filtering Pi contains the following mapping:
[] = []
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga
.(x1, x2) = .(x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x2)
REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x2, x5)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

REV_IN_GGA(L, S) → U3_GGA(L, S, head_in_ga(L))
U3_GGA(L, S, head_out_ga) → U4_GGA(S, tail_in_ga(L))
U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(X2)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(13) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA(L, S) → U3_GGA(L, S, head_in_ga(L)) we obtained the following new rules [LPAR04]:

REV_IN_GGA(z1, .(z0)) → U3_GGA(z1, .(z0), head_in_ga(z1))

(14) Obligation:

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

U3_GGA(L, S, head_out_ga) → U4_GGA(S, tail_in_ga(L))
U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA(z1, .(z0)) → U3_GGA(z1, .(z0), head_in_ga(z1))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(X2)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(15) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV_IN_GGA(L, S) → U3_GGA(L, S, head_in_ga(L)) at position [2] we obtained the following new rules [LPAR04]:

REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)

(16) Obligation:

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

U3_GGA(L, S, head_out_ga) → U4_GGA(S, tail_in_ga(L))
U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga
head_in_ga(.(X2)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(17) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(18) Obligation:

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

U3_GGA(L, S, head_out_ga) → U4_GGA(S, tail_in_ga(L))
U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(19) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

head_in_ga(x0)

(20) Obligation:

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

U3_GGA(L, S, head_out_ga) → U4_GGA(S, tail_in_ga(L))
U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U3_GGA(L, S, head_out_ga) → U4_GGA(S, tail_in_ga(L)) at position [1] we obtained the following new rules [LPAR04]:

U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U3_GGA(.(x0), y1, head_out_ga) → U4_GGA(y1, tail_out_ga(x0))

(22) Obligation:

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

U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)
U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U3_GGA(.(x0), y1, head_out_ga) → U4_GGA(y1, tail_out_ga(x0))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(23) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(24) Obligation:

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

U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)
U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U3_GGA(.(x0), y1, head_out_ga) → U4_GGA(y1, tail_out_ga(x0))

R is empty.
The set Q consists of the following terms:

tail_in_ga(x0)

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

(25) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

tail_in_ga(x0)

(26) Obligation:

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

U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)
U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U3_GGA(.(x0), y1, head_out_ga) → U4_GGA(y1, tail_out_ga(x0))

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U3_GGA(.(x0), y1, head_out_ga) → U4_GGA(y1, tail_out_ga(x0))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U4_GGA(x₁, x₂)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 1 ]

· x₂

POL(tail_out_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁

POL(REV_IN_GGA(x₁, x₂)) = 0 +
[ 1, 0 ]

· x₁ +
[ 0, 0 ]

· x₂

POL(.(x₁)) =
/ 1 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL([]) =
/ 0 \

\ 0 /

POL(U3_GGA(x₁, x₂, x₃)) = 0 +
[ 0, 1 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃

POL(head_out_ga) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

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

U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga)
U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))

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

(29) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(30) Obligation:

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

REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))

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

(31) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga) we obtained the following new rules [LPAR04]:

REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)

(32) Obligation:

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

U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)

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

(33) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GGA(S, tail_out_ga(T)) → REV_IN_GGA(T, .(S)) we obtained the following new rules [LPAR04]:

U4_GGA(z0, tail_out_ga([])) → REV_IN_GGA([], .(z0))

(34) Obligation:

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

REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga)
U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U4_GGA(z0, tail_out_ga([])) → REV_IN_GGA([], .(z0))

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

(35) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga) we obtained the following new rules [LPAR04]:

REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)

(36) Obligation:

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

U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U4_GGA(z0, tail_out_ga([])) → REV_IN_GGA([], .(z0))
REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)

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

(37) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga) we obtained the following new rules [LPAR04]:

REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)

(38) Obligation:

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

U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([]))
U4_GGA(z0, tail_out_ga([])) → REV_IN_GGA([], .(z0))
REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)

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

(39) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GGA([], y1, head_out_ga) → U4_GGA(y1, tail_out_ga([])) we obtained the following new rules [LPAR04]:

U3_GGA([], .(z0), head_out_ga) → U4_GGA(.(z0), tail_out_ga([]))

(40) Obligation:

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

U4_GGA(z0, tail_out_ga([])) → REV_IN_GGA([], .(z0))
REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)
U3_GGA([], .(z0), head_out_ga) → U4_GGA(.(z0), tail_out_ga([]))

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

(41) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GGA(z0, tail_out_ga([])) → REV_IN_GGA([], .(z0)) we obtained the following new rules [LPAR04]:

U4_GGA(.(z0), tail_out_ga([])) → REV_IN_GGA([], .(.(z0)))

(42) Obligation:

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

REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga)
U3_GGA([], .(z0), head_out_ga) → U4_GGA(.(z0), tail_out_ga([]))
U4_GGA(.(z0), tail_out_ga([])) → REV_IN_GGA([], .(.(z0)))

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

(43) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga) we obtained the following new rules [LPAR04]:

REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga)

(44) Obligation:

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

U3_GGA([], .(z0), head_out_ga) → U4_GGA(.(z0), tail_out_ga([]))
U4_GGA(.(z0), tail_out_ga([])) → REV_IN_GGA([], .(.(z0)))
REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga)

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

(45) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GGA([], .(z0), head_out_ga) → U4_GGA(.(z0), tail_out_ga([])) we obtained the following new rules [LPAR04]:

U3_GGA([], .(.(z0)), head_out_ga) → U4_GGA(.(.(z0)), tail_out_ga([]))

(46) Obligation:

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

U4_GGA(.(z0), tail_out_ga([])) → REV_IN_GGA([], .(.(z0)))
REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga)
U3_GGA([], .(.(z0)), head_out_ga) → U4_GGA(.(.(z0)), tail_out_ga([]))

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

(47) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = REV_IN_GGA([], .(.(z0))) evaluates to t =REV_IN_GGA([], .(.(.(z0))))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [z0 / .(z0)]
Semiunifier: [ ]

Rewriting sequence

REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga)
with rule REV_IN_GGA([], .(.(z0'))) → U3_GGA([], .(.(z0')), head_out_ga) at position [] and matcher [z0' / z0]

U3_GGA([], .(.(z0)), head_out_ga) → U4_GGA(.(.(z0)), tail_out_ga([]))
with rule U3_GGA([], .(.(z0')), head_out_ga) → U4_GGA(.(.(z0')), tail_out_ga([])) at position [] and matcher [z0' / z0]

U4_GGA(.(.(z0)), tail_out_ga([])) → REV_IN_GGA([], .(.(.(z0))))
with rule U4_GGA(.(z0), tail_out_ga([])) → REV_IN_GGA([], .(.(z0)))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(48) FALSE

(49) PrologToPiTRSProof (SOUND transformation)

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(50) Obligation:

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

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

(51) 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(L, R) → U1_GA(L, R, rev_in_gga(L, [], R))
REV_IN_GA(L, R) → REV_IN_GGA(L, [], R)
REV_IN_GGA([], Y, Z) → U2_GGA(Y, Z, eq_in_ga(Y, Z))
REV_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
REV_IN_GGA(L, S, R) → HEAD_IN_GA(L, X)
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U3_GGA(L, S, R, head_out_ga(L, X)) → TAIL_IN_GA(L, T)
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → U5_GGA(L, S, R, rev_in_gga(T, .(X, S), R))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2) = rev_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2)
[] = []
U2_gga(x1, x2, x3) = U2_gga(x1, x3)
eq_in_ga(x1, x2) = eq_in_ga(x1)
eq_out_ga(x1, x2) = eq_out_ga(x1, x2)
rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga(x1)
.(x1, x2) = .(x2)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
rev_out_ga(x1, x2) = rev_out_ga(x1, x2)
REV_IN_GA(x1, x2) = REV_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3) = U2_GGA(x1, x3)
EQ_IN_GA(x1, x2) = EQ_IN_GA(x1)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
HEAD_IN_GA(x1, x2) = HEAD_IN_GA(x1)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x5)
TAIL_IN_GA(x1, x2) = TAIL_IN_GA(x1)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)

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

(52) Obligation:

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

REV_IN_GA(L, R) → U1_GA(L, R, rev_in_gga(L, [], R))
REV_IN_GA(L, R) → REV_IN_GGA(L, [], R)
REV_IN_GGA([], Y, Z) → U2_GGA(Y, Z, eq_in_ga(Y, Z))
REV_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
REV_IN_GGA(L, S, R) → HEAD_IN_GA(L, X)
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U3_GGA(L, S, R, head_out_ga(L, X)) → TAIL_IN_GA(L, T)
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → U5_GGA(L, S, R, rev_in_gga(T, .(X, S), R))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

(53) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.

(54) Obligation:

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

REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

rev_in_ga(L, R) → U1_ga(L, R, rev_in_gga(L, [], R))
rev_in_gga([], Y, Z) → U2_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U2_gga(Y, Z, eq_out_ga(Y, Z)) → rev_out_gga([], Y, Z)
rev_in_gga(L, S, R) → U3_gga(L, S, R, head_in_ga(L, X))
head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
U3_gga(L, S, R, head_out_ga(L, X)) → U4_gga(L, S, R, X, tail_in_ga(L, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)
U4_gga(L, S, R, X, tail_out_ga(L, T)) → U5_gga(L, S, R, rev_in_gga(T, .(X, S), R))
U5_gga(L, S, R, rev_out_gga(T, .(X, S), R)) → rev_out_gga(L, S, R)
U1_ga(L, R, rev_out_gga(L, [], R)) → rev_out_ga(L, R)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2) = rev_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x1, x3)
rev_in_gga(x1, x2, x3) = rev_in_gga(x1, x2)
[] = []
U2_gga(x1, x2, x3) = U2_gga(x1, x3)
eq_in_ga(x1, x2) = eq_in_ga(x1)
eq_out_ga(x1, x2) = eq_out_ga(x1, x2)
rev_out_gga(x1, x2, x3) = rev_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga(x1)
.(x1, x2) = .(x2)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x5)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
rev_out_ga(x1, x2) = rev_out_ga(x1, x2)
REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x5)

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

(55) UsableRulesProof (EQUIVALENT transformation)

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

(56) Obligation:

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

REV_IN_GGA(L, S, R) → U3_GGA(L, S, R, head_in_ga(L, X))
U3_GGA(L, S, R, head_out_ga(L, X)) → U4_GGA(L, S, R, X, tail_in_ga(L, T))
U4_GGA(L, S, R, X, tail_out_ga(L, T)) → REV_IN_GGA(T, .(X, S), R)

The TRS R consists of the following rules:

head_in_ga([], X1) → head_out_ga([], X1)
head_in_ga(.(X, X2), X) → head_out_ga(.(X, X2), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X3, Xs), Xs) → tail_out_ga(.(X3, Xs), Xs)

The argument filtering Pi contains the following mapping:
[] = []
head_in_ga(x1, x2) = head_in_ga(x1)
head_out_ga(x1, x2) = head_out_ga(x1)
.(x1, x2) = .(x2)
tail_in_ga(x1, x2) = tail_in_ga(x1)
tail_out_ga(x1, x2) = tail_out_ga(x1, x2)
REV_IN_GGA(x1, x2, x3) = REV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x5)

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

(57) PiDPToQDPProof (SOUND transformation)

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

(58) Obligation:

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

REV_IN_GGA(L, S) → U3_GGA(L, S, head_in_ga(L))
U3_GGA(L, S, head_out_ga(L)) → U4_GGA(L, S, tail_in_ga(L))
U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(X2)) → head_out_ga(.(X2))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(59) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA(L, S) → U3_GGA(L, S, head_in_ga(L)) we obtained the following new rules [LPAR04]:

REV_IN_GGA(z2, .(z1)) → U3_GGA(z2, .(z1), head_in_ga(z2))

(60) Obligation:

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

U3_GGA(L, S, head_out_ga(L)) → U4_GGA(L, S, tail_in_ga(L))
U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA(z2, .(z1)) → U3_GGA(z2, .(z1), head_in_ga(z2))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(X2)) → head_out_ga(.(X2))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(61) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule REV_IN_GGA(L, S) → U3_GGA(L, S, head_in_ga(L)) at position [2] we obtained the following new rules [LPAR04]:

REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))

(62) Obligation:

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

U3_GGA(L, S, head_out_ga(L)) → U4_GGA(L, S, tail_in_ga(L))
U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))

The TRS R consists of the following rules:

head_in_ga([]) → head_out_ga([])
head_in_ga(.(X2)) → head_out_ga(.(X2))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(63) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(64) Obligation:

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

U3_GGA(L, S, head_out_ga(L)) → U4_GGA(L, S, tail_in_ga(L))
U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

head_in_ga(x0)
tail_in_ga(x0)

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

(65) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

head_in_ga(x0)

(66) Obligation:

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

U3_GGA(L, S, head_out_ga(L)) → U4_GGA(L, S, tail_in_ga(L))
U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(67) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U3_GGA(L, S, head_out_ga(L)) → U4_GGA(L, S, tail_in_ga(L)) at position [2] we obtained the following new rules [LPAR04]:

U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U3_GGA(.(x0), y1, head_out_ga(.(x0))) → U4_GGA(.(x0), y1, tail_out_ga(.(x0), x0))

(68) Obligation:

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

U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))
U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U3_GGA(.(x0), y1, head_out_ga(.(x0))) → U4_GGA(.(x0), y1, tail_out_ga(.(x0), x0))

The TRS R consists of the following rules:

tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)

The set Q consists of the following terms:

tail_in_ga(x0)

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

(69) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(70) Obligation:

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

U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))
U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U3_GGA(.(x0), y1, head_out_ga(.(x0))) → U4_GGA(.(x0), y1, tail_out_ga(.(x0), x0))

R is empty.
The set Q consists of the following terms:

tail_in_ga(x0)

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

(71) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

tail_in_ga(x0)

(72) Obligation:

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

U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))
U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U3_GGA(.(x0), y1, head_out_ga(.(x0))) → U4_GGA(.(x0), y1, tail_out_ga(.(x0), x0))

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

(73) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GGA(L, S, tail_out_ga(L, T)) → REV_IN_GGA(T, .(S)) we obtained the following new rules [LPAR04]:

U4_GGA([], z0, tail_out_ga([], [])) → REV_IN_GGA([], .(z0))
U4_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → REV_IN_GGA(z0, .(z1))

(74) Obligation:

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

REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))
U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U3_GGA(.(x0), y1, head_out_ga(.(x0))) → U4_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
U4_GGA([], z0, tail_out_ga([], [])) → REV_IN_GGA([], .(z0))
U4_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → REV_IN_GGA(z0, .(z1))

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

(75) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(76) Complex Obligation (AND)

(77) Obligation:

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

U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U4_GGA([], z0, tail_out_ga([], [])) → REV_IN_GGA([], .(z0))
REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([]))

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

(78) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([])) we obtained the following new rules [LPAR04]:

REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga([]))

(79) Obligation:

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

U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U4_GGA([], z0, tail_out_ga([], [])) → REV_IN_GGA([], .(z0))
REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga([]))

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

(80) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA([], y1) → U3_GGA([], y1, head_out_ga([])) we obtained the following new rules [LPAR04]:

REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga([]))

(81) Obligation:

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

U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], []))
U4_GGA([], z0, tail_out_ga([], [])) → REV_IN_GGA([], .(z0))
REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga([]))

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

(82) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GGA([], y1, head_out_ga([])) → U4_GGA([], y1, tail_out_ga([], [])) we obtained the following new rules [LPAR04]:

U3_GGA([], .(z0), head_out_ga([])) → U4_GGA([], .(z0), tail_out_ga([], []))

(83) Obligation:

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

U4_GGA([], z0, tail_out_ga([], [])) → REV_IN_GGA([], .(z0))
REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga([]))
U3_GGA([], .(z0), head_out_ga([])) → U4_GGA([], .(z0), tail_out_ga([], []))

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

(84) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GGA([], z0, tail_out_ga([], [])) → REV_IN_GGA([], .(z0)) we obtained the following new rules [LPAR04]:

U4_GGA([], .(z0), tail_out_ga([], [])) → REV_IN_GGA([], .(.(z0)))

(85) Obligation:

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

REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga([]))
U3_GGA([], .(z0), head_out_ga([])) → U4_GGA([], .(z0), tail_out_ga([], []))
U4_GGA([], .(z0), tail_out_ga([], [])) → REV_IN_GGA([], .(.(z0)))

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

(86) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REV_IN_GGA([], .(z0)) → U3_GGA([], .(z0), head_out_ga([])) we obtained the following new rules [LPAR04]:

REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga([]))

(87) Obligation:

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

U3_GGA([], .(z0), head_out_ga([])) → U4_GGA([], .(z0), tail_out_ga([], []))
U4_GGA([], .(z0), tail_out_ga([], [])) → REV_IN_GGA([], .(.(z0)))
REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga([]))

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

(88) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GGA([], .(z0), head_out_ga([])) → U4_GGA([], .(z0), tail_out_ga([], [])) we obtained the following new rules [LPAR04]:

U3_GGA([], .(.(z0)), head_out_ga([])) → U4_GGA([], .(.(z0)), tail_out_ga([], []))

(89) Obligation:

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

U4_GGA([], .(z0), tail_out_ga([], [])) → REV_IN_GGA([], .(.(z0)))
REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga([]))
U3_GGA([], .(.(z0)), head_out_ga([])) → U4_GGA([], .(.(z0)), tail_out_ga([], []))

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

(90) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [z0 / .(z0)]

Rewriting sequence

REV_IN_GGA([], .(.(z0))) → U3_GGA([], .(.(z0)), head_out_ga([]))
with rule REV_IN_GGA([], .(.(z0'))) → U3_GGA([], .(.(z0')), head_out_ga([])) at position [] and matcher [z0' / z0]

U3_GGA([], .(.(z0)), head_out_ga([])) → U4_GGA([], .(.(z0)), tail_out_ga([], []))
with rule U3_GGA([], .(.(z0')), head_out_ga([])) → U4_GGA([], .(.(z0')), tail_out_ga([], [])) at position [] and matcher [z0' / z0]

U4_GGA([], .(.(z0)), tail_out_ga([], [])) → REV_IN_GGA([], .(.(.(z0))))
with rule U4_GGA([], .(z0), tail_out_ga([], [])) → REV_IN_GGA([], .(.(z0)))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(91) FALSE

(92) Obligation:

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

REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))
U3_GGA(.(x0), y1, head_out_ga(.(x0))) → U4_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
U4_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → REV_IN_GGA(z0, .(z1))

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

(93) 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:

U3_GGA(.(x0), y1, head_out_ga(.(x0))) → U4_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
U4_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → REV_IN_GGA(z0, .(z1))
The graph contains the following edges 1 > 1, 3 > 1
REV_IN_GGA(.(x0), y1) → U3_GGA(.(x0), y1, head_out_ga(.(x0)))
The graph contains the following edges 1 >= 1, 2 >= 2