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


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WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 197 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsTAProof [FINISHED, 393 ms]
(8) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))

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(cons(x_1)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_nats(x_1)) -> nats(encArg(x_1))
   encArg(cons_zprimes) -> zprimes
   encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_nats(x_1) -> nats(encArg(x_1))
   encode_zprimes -> zprimes


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

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


The TRS R consists of the following rules:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))

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

   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_nats(x_1)) -> nats(encArg(x_1))
   encArg(cons_zprimes) -> zprimes
   encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_nats(x_1) -> nats(encArg(x_1))
   encode_zprimes -> zprimes

Rewrite Strategy: INNERMOST
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(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(1, n^1).


The TRS R consists of the following rules:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))

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

   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_nats(x_1)) -> nats(encArg(x_1))
   encArg(cons_zprimes) -> zprimes
   encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_nats(x_1) -> nats(encArg(x_1))
   encode_zprimes -> zprimes

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

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_filter(x_1, x_2, x_3)) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_sieve(x_1)) -> sieve(encArg(x_1))
   encArg(cons_nats(x_1)) -> nats(encArg(x_1))
   encArg(cons_zprimes) -> zprimes
   encode_filter(x_1, x_2, x_3) -> filter(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_sieve(x_1) -> sieve(encArg(x_1))
   encode_nats(x_1) -> nats(encArg(x_1))
   encode_zprimes -> zprimes

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

(7) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
transitions: 
cons0(0) -> 0
00() -> 0
s0(0) -> 0
cons_filter0(0, 0, 0) -> 0
cons_sieve0(0) -> 0
cons_nats0(0) -> 0
cons_zprimes0() -> 0
filter0(0, 0, 0) -> 1
sieve0(0) -> 2
nats0(0) -> 3
zprimes0() -> 4
encArg0(0) -> 5
encode_filter0(0, 0, 0) -> 6
encode_cons0(0) -> 7
encode_00() -> 8
encode_s0(0) -> 9
encode_sieve0(0) -> 10
encode_nats0(0) -> 11
encode_zprimes0() -> 12
01() -> 13
cons1(13) -> 1
cons1(0) -> 1
cons1(13) -> 2
s1(0) -> 14
cons1(14) -> 2
cons1(0) -> 3
01() -> 18
s1(18) -> 17
s1(17) -> 16
nats1(16) -> 15
sieve1(15) -> 4
encArg1(0) -> 19
cons1(19) -> 5
01() -> 5
encArg1(0) -> 20
s1(20) -> 5
encArg1(0) -> 21
encArg1(0) -> 22
encArg1(0) -> 23
filter1(21, 22, 23) -> 5
encArg1(0) -> 24
sieve1(24) -> 5
encArg1(0) -> 25
nats1(25) -> 5
zprimes1() -> 5
filter1(21, 22, 23) -> 6
cons1(19) -> 7
01() -> 8
s1(20) -> 9
sieve1(24) -> 10
nats1(25) -> 11
zprimes1() -> 12
cons2(25) -> 5
cons2(25) -> 11
cons2(16) -> 15
02() -> 29
s2(29) -> 28
s2(28) -> 27
nats2(27) -> 26
sieve2(26) -> 5
sieve2(26) -> 12
cons1(19) -> 19
cons1(19) -> 20
cons1(19) -> 21
cons1(19) -> 22
cons1(19) -> 23
cons1(19) -> 24
cons1(19) -> 25
01() -> 19
01() -> 20
01() -> 21
01() -> 22
01() -> 23
01() -> 24
01() -> 25
s1(20) -> 19
s1(20) -> 20
s1(20) -> 21
s1(20) -> 22
s1(20) -> 23
s1(20) -> 24
s1(20) -> 25
filter1(21, 22, 23) -> 19
filter1(21, 22, 23) -> 20
filter1(21, 22, 23) -> 21
filter1(21, 22, 23) -> 22
filter1(21, 22, 23) -> 23
filter1(21, 22, 23) -> 24
filter1(21, 22, 23) -> 25
sieve1(24) -> 19
sieve1(24) -> 20
sieve1(24) -> 21
sieve1(24) -> 22
sieve1(24) -> 23
sieve1(24) -> 24
sieve1(24) -> 25
nats1(25) -> 19
nats1(25) -> 20
nats1(25) -> 21
nats1(25) -> 22
nats1(25) -> 23
nats1(25) -> 24
nats1(25) -> 25
zprimes1() -> 19
zprimes1() -> 20
zprimes1() -> 21
zprimes1() -> 22
zprimes1() -> 23
zprimes1() -> 24
zprimes1() -> 25
02() -> 30
cons2(30) -> 5
cons2(30) -> 6
cons2(30) -> 19
cons2(30) -> 20
cons2(30) -> 21
cons2(30) -> 22
cons2(30) -> 23
cons2(30) -> 24
cons2(30) -> 25
cons2(19) -> 5
cons2(19) -> 6
cons2(19) -> 19
cons2(19) -> 20
cons2(19) -> 21
cons2(19) -> 22
cons2(19) -> 23
cons2(19) -> 24
cons2(19) -> 25
cons2(30) -> 10
s2(20) -> 31
cons2(31) -> 5
cons2(31) -> 10
cons2(31) -> 19
cons2(31) -> 20
cons2(31) -> 21
cons2(31) -> 22
cons2(31) -> 23
cons2(31) -> 24
cons2(31) -> 25
cons2(25) -> 19
cons2(25) -> 20
cons2(25) -> 21
cons2(25) -> 22
cons2(25) -> 23
cons2(25) -> 24
cons2(25) -> 25
sieve2(26) -> 19
sieve2(26) -> 20
sieve2(26) -> 21
sieve2(26) -> 22
sieve2(26) -> 23
sieve2(26) -> 24
sieve2(26) -> 25
s2(17) -> 31
cons2(31) -> 4
cons3(27) -> 26
cons2(31) -> 6
cons2(25) -> 6
s3(28) -> 32
cons3(32) -> 5
cons3(32) -> 12
cons3(32) -> 19
cons3(32) -> 20
cons3(32) -> 21
cons3(32) -> 22
cons3(32) -> 23
cons3(32) -> 24
cons3(32) -> 25
cons2(32) -> 5
cons2(32) -> 6
cons2(32) -> 19
cons2(32) -> 20
cons2(32) -> 21
cons2(32) -> 22
cons2(32) -> 23
cons2(32) -> 24
cons2(32) -> 25
s2(28) -> 31

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(8)
BOUNDS(1, n^1)