/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) 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), 188 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsTAProof [FINISHED, 97 ms]
(8) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)
   from(X) -> cons(X)

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

(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(nil) -> nil
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))


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

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)
   from(X) -> cons(X)

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

   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)
   from(X) -> cons(X)

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

   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)
   from(X) -> cons(X)
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s(x_1) -> s(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))

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

(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 2. 

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]
transitions: 
00() -> 0
nil0() -> 0
s0(0) -> 0
cons0(0) -> 0
cons_first0(0, 0) -> 0
cons_from0(0) -> 0
first0(0, 0) -> 1
from0(0) -> 2
encArg0(0) -> 3
encode_first0(0, 0) -> 4
encode_00() -> 5
encode_nil0() -> 6
encode_s0(0) -> 7
encode_cons0(0) -> 8
encode_from0(0) -> 9
nil1() -> 1
cons1(0) -> 1
cons1(0) -> 2
01() -> 3
nil1() -> 3
encArg1(0) -> 10
s1(10) -> 3
encArg1(0) -> 11
cons1(11) -> 3
encArg1(0) -> 12
encArg1(0) -> 13
first1(12, 13) -> 3
encArg1(0) -> 14
from1(14) -> 3
first1(12, 13) -> 4
01() -> 5
nil1() -> 6
s1(10) -> 7
cons1(11) -> 8
from1(14) -> 9
cons2(14) -> 3
cons2(14) -> 9
01() -> 10
01() -> 11
01() -> 12
01() -> 13
01() -> 14
nil1() -> 10
nil1() -> 11
nil1() -> 12
nil1() -> 13
nil1() -> 14
s1(10) -> 10
s1(10) -> 11
s1(10) -> 12
s1(10) -> 13
s1(10) -> 14
cons1(11) -> 10
cons1(11) -> 11
cons1(11) -> 12
cons1(11) -> 13
cons1(11) -> 14
first1(12, 13) -> 10
first1(12, 13) -> 11
first1(12, 13) -> 12
first1(12, 13) -> 13
first1(12, 13) -> 14
from1(14) -> 10
from1(14) -> 11
from1(14) -> 12
from1(14) -> 13
from1(14) -> 14
nil2() -> 3
nil2() -> 4
nil2() -> 10
nil2() -> 11
nil2() -> 12
nil2() -> 13
nil2() -> 14
cons2(11) -> 3
cons2(11) -> 4
cons2(11) -> 10
cons2(11) -> 11
cons2(11) -> 12
cons2(11) -> 13
cons2(11) -> 14
cons2(14) -> 10
cons2(14) -> 11
cons2(14) -> 12
cons2(14) -> 13
cons2(14) -> 14
cons2(14) -> 4

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(8)
BOUNDS(1, n^1)