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


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WORST_CASE(Omega(n^1), O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 164 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsTAProof [FINISHED, 1013 ms]
    (8) BOUNDS(1, n^1)
(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(10) TRS for Loop Detection
    (11) DecreasingLoopProof [LOWER BOUND(ID), 14 ms]
    (12) BEST
        (13) proven lower bound
            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
            (15) BOUNDS(n^1, INF)
        (16) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 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(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))


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

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


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(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, n^1).


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)
   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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 4. 

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: 
true0() -> 0
false0() -> 0
00() -> 0
s0(0) -> 0
cons_not0(0) -> 0
cons_evenodd0(0, 0) -> 0
not0(0) -> 1
evenodd0(0, 0) -> 2
encArg0(0) -> 3
encode_not0(0) -> 4
encode_true0() -> 5
encode_false0() -> 6
encode_evenodd0(0, 0) -> 7
encode_00() -> 8
encode_s0(0) -> 9
false1() -> 1
true1() -> 1
01() -> 12
s1(12) -> 11
evenodd1(0, 11) -> 10
not1(10) -> 2
false1() -> 2
01() -> 13
evenodd1(0, 13) -> 2
true1() -> 3
false1() -> 3
01() -> 3
encArg1(0) -> 14
s1(14) -> 3
encArg1(0) -> 15
not1(15) -> 3
encArg1(0) -> 16
encArg1(0) -> 17
evenodd1(16, 17) -> 3
not1(15) -> 4
true1() -> 5
false1() -> 6
evenodd1(16, 17) -> 7
01() -> 8
s1(14) -> 9
02() -> 20
s2(20) -> 19
evenodd2(0, 19) -> 18
not2(18) -> 2
false1() -> 10
evenodd1(0, 13) -> 10
true1() -> 14
true1() -> 15
true1() -> 16
true1() -> 17
false1() -> 14
false1() -> 15
false1() -> 16
false1() -> 17
01() -> 14
01() -> 15
01() -> 16
01() -> 17
s1(14) -> 14
s1(14) -> 15
s1(14) -> 16
s1(14) -> 17
not1(15) -> 14
not1(15) -> 15
not1(15) -> 16
not1(15) -> 17
evenodd1(16, 17) -> 14
evenodd1(16, 17) -> 15
evenodd1(16, 17) -> 16
evenodd1(16, 17) -> 17
false2() -> 3
false2() -> 4
false2() -> 14
false2() -> 15
false2() -> 16
false2() -> 17
true2() -> 2
true2() -> 3
true2() -> 4
true2() -> 14
true2() -> 15
true2() -> 16
true2() -> 17
evenodd2(16, 19) -> 18
not2(18) -> 3
not2(18) -> 7
not2(18) -> 10
not2(18) -> 14
not2(18) -> 15
not2(18) -> 16
not2(18) -> 17
false2() -> 7
02() -> 21
evenodd2(14, 21) -> 3
evenodd2(14, 21) -> 7
evenodd2(14, 21) -> 14
evenodd2(14, 21) -> 15
evenodd2(14, 21) -> 16
evenodd2(14, 21) -> 17
false1() -> 18
evenodd1(0, 13) -> 18
not2(18) -> 18
true2() -> 7
true2() -> 10
03() -> 24
s3(24) -> 23
evenodd3(14, 23) -> 22
not3(22) -> 3
not3(22) -> 7
not3(22) -> 14
not3(22) -> 15
not3(22) -> 16
not3(22) -> 17
false2() -> 18
evenodd2(14, 21) -> 18
true2() -> 18
not3(22) -> 18
false2() -> 2
true3() -> 2
true3() -> 3
true3() -> 7
true3() -> 10
true3() -> 14
true3() -> 15
true3() -> 16
true3() -> 17
true3() -> 18
false2() -> 22
evenodd2(14, 21) -> 22
not3(22) -> 22
false3() -> 2
false3() -> 3
false3() -> 7
false3() -> 10
false3() -> 14
false3() -> 15
false3() -> 16
false3() -> 17
false3() -> 18
true3() -> 22
false4() -> 3
false4() -> 7
false4() -> 14
false4() -> 15
false4() -> 16
false4() -> 17
false4() -> 18
false4() -> 22
true4() -> 3
true4() -> 7
true4() -> 14
true4() -> 15
true4() -> 16
true4() -> 17
true4() -> 18
true4() -> 22

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

(8)
BOUNDS(1, n^1)

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

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

(10)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

The rewrite sequence

evenodd(s(x1_0), 0) ->^+ not(evenodd(x1_0, 0))

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

The pumping substitution is [x1_0 / s(x1_0)].

The result substitution is [ ].




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

(12)
Complex Obligation (BEST)

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

(13)
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, n^1).


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: INNERMOST