Left Termination proof of /tmp/tmpqDW1Vd/naive

Left Termination of the query pattern reverse_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:

app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
reverse(.(X, Xs), Ys) :- ','(reverse(Xs, Zs), app(Zs, .(X, []), Ys)).
reverse([], []).

Queries:

reverse(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:
reverse_in: (b,f)
app_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

(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:

REVERSE_IN_GA(.(X, Xs), Ys) → U2_GA(X, Xs, Ys, reverse_in_ga(Xs, Zs))
REVERSE_IN_GA(.(X, Xs), Ys) → REVERSE_IN_GA(Xs, Zs)
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → APP_IN_GGA(Zs, .(X, []), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2) = reverse_in_ga(x1)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4)
[] = []
reverse_out_ga(x1, x2) = reverse_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x1, x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3)
REVERSE_IN_GA(x1, x2) = REVERSE_IN_GA(x1)
U2_GA(x1, x2, x3, x4) = U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x1, x2, x4)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5)

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

(4) Obligation:

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

REVERSE_IN_GA(.(X, Xs), Ys) → U2_GA(X, Xs, Ys, reverse_in_ga(Xs, Zs))
REVERSE_IN_GA(.(X, Xs), Ys) → REVERSE_IN_GA(Xs, Zs)
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → APP_IN_GGA(Zs, .(X, []), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

(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:

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

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2) = reverse_in_ga(x1)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4)
[] = []
reverse_out_ga(x1, x2) = reverse_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x1, x2, x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x1, x2, x3)
APP_IN_GGA(x1, x2, x3) = APP_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:

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

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

APP_IN_GGA(.(X, Xs), Ys) → APP_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:

APP_IN_GGA(.(X, Xs), Ys) → APP_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:

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

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

(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:

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

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

REVERSE_IN_GA(.(X, Xs)) → REVERSE_IN_GA(Xs)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

REVERSE_IN_GA(.(X, Xs)) → REVERSE_IN_GA(Xs)
The graph contains the following edges 1 > 1

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

(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:

REVERSE_IN_GA(.(X, Xs), Ys) → U2_GA(X, Xs, Ys, reverse_in_ga(Xs, Zs))
REVERSE_IN_GA(.(X, Xs), Ys) → REVERSE_IN_GA(Xs, Zs)
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → APP_IN_GGA(Zs, .(X, []), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2) = reverse_in_ga(x1)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4) = U2_ga(x1, x4)
[] = []
reverse_out_ga(x1, x2) = reverse_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
REVERSE_IN_GA(x1, x2) = REVERSE_IN_GA(x1)
U2_GA(x1, x2, x3, x4) = U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x5)

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

(24) Obligation:

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

REVERSE_IN_GA(.(X, Xs), Ys) → U2_GA(X, Xs, Ys, reverse_in_ga(Xs, Zs))
REVERSE_IN_GA(.(X, Xs), Ys) → REVERSE_IN_GA(Xs, Zs)
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
U2_GA(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → APP_IN_GGA(Zs, .(X, []), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) Complex Obligation (AND)

(27) Obligation:

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

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

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
reverse_in_ga(x1, x2) = reverse_in_ga(x1)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4) = U2_ga(x1, x4)
[] = []
reverse_out_ga(x1, x2) = reverse_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x5)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
APP_IN_GGA(x1, x2, x3) = APP_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:

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

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

APP_IN_GGA(.(X, Xs), Ys) → APP_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:

APP_IN_GGA(.(X, Xs), Ys) → APP_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:

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

The TRS R consists of the following rules:

reverse_in_ga(.(X, Xs), Ys) → U2_ga(X, Xs, Ys, reverse_in_ga(Xs, Zs))
reverse_in_ga([], []) → reverse_out_ga([], [])
U2_ga(X, Xs, Ys, reverse_out_ga(Xs, Zs)) → U3_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
app_in_gga([], Ys, Ys) → app_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U3_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → reverse_out_ga(.(X, Xs), Ys)

(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:

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

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

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:

REVERSE_IN_GA(.(X, Xs)) → REVERSE_IN_GA(Xs)

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