/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 469 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   primes -> sieve(from(s(s(0))))
   from(X) -> cons(X, n__from(n__s(X)))
   head(cons(X, Y)) -> X
   tail(cons(X, Y)) -> activate(Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   filter(X1, X2) -> n__filter(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   sieve(X) -> n__sieve(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__sieve(X)) -> sieve(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2))
   encArg(n__filter(x_1, x_2)) -> n__filter(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__sieve(x_1)) -> n__sieve(encArg(x_1))
   encArg(cons_primes) -> primes
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_primes -> primes
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false
   encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2))
   encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2))
   encode_n__filter(x_1, x_2) -> n__filter(encArg(x_1), encArg(x_2))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__sieve(x_1) -> n__sieve(encArg(x_1))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   primes -> sieve(from(s(s(0))))
   from(X) -> cons(X, n__from(n__s(X)))
   head(cons(X, Y)) -> X
   tail(cons(X, Y)) -> activate(Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   filter(X1, X2) -> n__filter(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   sieve(X) -> n__sieve(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__sieve(X)) -> sieve(activate(X))
   activate(X) -> X

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2))
   encArg(n__filter(x_1, x_2)) -> n__filter(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__sieve(x_1)) -> n__sieve(encArg(x_1))
   encArg(cons_primes) -> primes
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_primes -> primes
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false
   encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2))
   encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2))
   encode_n__filter(x_1, x_2) -> n__filter(encArg(x_1), encArg(x_2))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__sieve(x_1) -> n__sieve(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   primes -> sieve(from(s(s(0))))
   from(X) -> cons(X, n__from(n__s(X)))
   head(cons(X, Y)) -> X
   tail(cons(X, Y)) -> activate(Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   filter(X1, X2) -> n__filter(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   sieve(X) -> n__sieve(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__sieve(X)) -> sieve(activate(X))
   activate(X) -> X

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2))
   encArg(n__filter(x_1, x_2)) -> n__filter(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__sieve(x_1)) -> n__sieve(encArg(x_1))
   encArg(cons_primes) -> primes
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_primes -> primes
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false
   encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2))
   encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2))
   encode_n__filter(x_1, x_2) -> n__filter(encArg(x_1), encArg(x_2))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__sieve(x_1) -> n__sieve(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   primes -> sieve(from(s(s(0))))
   from(X) -> cons(X, n__from(n__s(X)))
   head(cons(X, Y)) -> X
   tail(cons(X, Y)) -> activate(Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   filter(X1, X2) -> n__filter(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   sieve(X) -> n__sieve(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__sieve(X)) -> sieve(activate(X))
   activate(X) -> X

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2))
   encArg(n__filter(x_1, x_2)) -> n__filter(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__sieve(x_1)) -> n__sieve(encArg(x_1))
   encArg(cons_primes) -> primes
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_primes -> primes
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false
   encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2))
   encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2))
   encode_n__filter(x_1, x_2) -> n__filter(encArg(x_1), encArg(x_2))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__sieve(x_1) -> n__sieve(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

activate(n__s(X)) ->^+ s(activate(X))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X / n__s(X)].

The result substitution is [ ].




----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   primes -> sieve(from(s(s(0))))
   from(X) -> cons(X, n__from(n__s(X)))
   head(cons(X, Y)) -> X
   tail(cons(X, Y)) -> activate(Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   filter(X1, X2) -> n__filter(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   sieve(X) -> n__sieve(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__sieve(X)) -> sieve(activate(X))
   activate(X) -> X

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2))
   encArg(n__filter(x_1, x_2)) -> n__filter(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__sieve(x_1)) -> n__sieve(encArg(x_1))
   encArg(cons_primes) -> primes
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_primes -> primes
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false
   encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2))
   encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2))
   encode_n__filter(x_1, x_2) -> n__filter(encArg(x_1), encArg(x_2))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__sieve(x_1) -> n__sieve(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   primes -> sieve(from(s(s(0))))
   from(X) -> cons(X, n__from(n__s(X)))
   head(cons(X, Y)) -> X
   tail(cons(X, Y)) -> activate(Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y))))
   sieve(cons(X, Y)) -> cons(X, n__filter(X, n__sieve(activate(Y))))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   filter(X1, X2) -> n__filter(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   sieve(X) -> n__sieve(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__filter(X1, X2)) -> filter(activate(X1), activate(X2))
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__sieve(X)) -> sieve(activate(X))
   activate(X) -> X

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(n__from(x_1)) -> n__from(encArg(x_1))
   encArg(n__s(x_1)) -> n__s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(divides(x_1, x_2)) -> divides(encArg(x_1), encArg(x_2))
   encArg(n__filter(x_1, x_2)) -> n__filter(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__sieve(x_1)) -> n__sieve(encArg(x_1))
   encArg(cons_primes) -> primes
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_filter(x_1, x_2)) -> filter(encArg(x_1), encArg(x_2))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_primes -> primes
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_n__from(x_1) -> n__from(encArg(x_1))
   encode_n__s(x_1) -> n__s(encArg(x_1))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_true -> true
   encode_false -> false
   encode_filter(x_1, x_2) -> filter(encArg(x_1), encArg(x_2))
   encode_divides(x_1, x_2) -> divides(encArg(x_1), encArg(x_2))
   encode_n__filter(x_1, x_2) -> n__filter(encArg(x_1), encArg(x_2))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__sieve(x_1) -> n__sieve(encArg(x_1))

Rewrite Strategy: INNERMOST