/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 167 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsTAProof [FINISHED, 920 ms]
    (8) BOUNDS(1, n^1)
(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(10) TRS for Loop Detection
    (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (12) BEST
        (13) proven lower bound
            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
            (15) BOUNDS(n^1, INF)
        (16) TRS for Loop Detection


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(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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))


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

(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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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]
transitions: 
00() -> 0
s0(0) -> 0
*0(0, 0) -> 0
+0(0, 0) -> 0
cons_f0(0) -> 0
f0(0) -> 1
encArg0(0) -> 2
encode_f0(0) -> 3
encode_00() -> 4
encode_s0(0) -> 5
encode_*0(0, 0) -> 6
encode_+0(0, 0) -> 7
01() -> 8
s1(8) -> 1
s1(8) -> 9
s1(9) -> 1
s1(9) -> 10
01() -> 12
f1(12) -> 11
*1(10, 11) -> 1
s1(9) -> 13
f1(0) -> 14
+1(13, 14) -> 1
f1(0) -> 15
f1(0) -> 16
*1(15, 16) -> 1
01() -> 2
encArg1(0) -> 17
s1(17) -> 2
encArg1(0) -> 18
encArg1(0) -> 19
*1(18, 19) -> 2
encArg1(0) -> 20
encArg1(0) -> 21
+1(20, 21) -> 2
encArg1(0) -> 22
f1(22) -> 2
f1(22) -> 3
01() -> 4
s1(17) -> 5
*1(18, 19) -> 6
+1(20, 21) -> 7
s1(8) -> 14
s1(8) -> 15
s1(8) -> 16
02() -> 23
s2(23) -> 11
s1(9) -> 14
s1(9) -> 15
s1(9) -> 16
*1(10, 11) -> 14
*1(10, 11) -> 15
*1(10, 11) -> 16
+1(13, 14) -> 14
+1(13, 14) -> 15
+1(13, 14) -> 16
*1(15, 16) -> 14
*1(15, 16) -> 15
*1(15, 16) -> 16
01() -> 17
01() -> 18
01() -> 19
01() -> 20
01() -> 21
01() -> 22
s1(17) -> 17
s1(17) -> 18
s1(17) -> 19
s1(17) -> 20
s1(17) -> 21
s1(17) -> 22
*1(18, 19) -> 17
*1(18, 19) -> 18
*1(18, 19) -> 19
*1(18, 19) -> 20
*1(18, 19) -> 21
*1(18, 19) -> 22
+1(20, 21) -> 17
+1(20, 21) -> 18
+1(20, 21) -> 19
+1(20, 21) -> 20
+1(20, 21) -> 21
+1(20, 21) -> 22
f1(22) -> 17
f1(22) -> 18
f1(22) -> 19
f1(22) -> 20
f1(22) -> 21
f1(22) -> 22
s2(23) -> 2
s2(23) -> 3
s2(23) -> 17
s2(23) -> 18
s2(23) -> 19
s2(23) -> 20
s2(23) -> 21
s2(23) -> 22
s2(23) -> 24
s2(24) -> 2
s2(24) -> 3
s2(24) -> 17
s2(24) -> 18
s2(24) -> 19
s2(24) -> 20
s2(24) -> 21
s2(24) -> 22
s2(24) -> 25
02() -> 27
f2(27) -> 26
*2(25, 26) -> 2
*2(25, 26) -> 3
*2(25, 26) -> 17
*2(25, 26) -> 18
*2(25, 26) -> 19
*2(25, 26) -> 20
*2(25, 26) -> 21
*2(25, 26) -> 22
s2(24) -> 28
f2(20) -> 29
+2(28, 29) -> 2
+2(28, 29) -> 3
+2(28, 29) -> 17
+2(28, 29) -> 18
+2(28, 29) -> 19
+2(28, 29) -> 20
+2(28, 29) -> 21
+2(28, 29) -> 22
f2(20) -> 30
f2(21) -> 31
*2(30, 31) -> 2
*2(30, 31) -> 3
*2(30, 31) -> 17
*2(30, 31) -> 18
*2(30, 31) -> 19
*2(30, 31) -> 20
*2(30, 31) -> 21
*2(30, 31) -> 22
s2(23) -> 29
s2(23) -> 30
s2(23) -> 31
03() -> 32
s3(32) -> 26
s2(24) -> 29
s2(24) -> 30
s2(24) -> 31
s3(32) -> 33
s3(33) -> 29
s3(33) -> 30
s3(33) -> 31
*2(25, 26) -> 29
*2(25, 26) -> 30
*2(25, 26) -> 31
s3(33) -> 34
03() -> 36
f3(36) -> 35
*3(34, 35) -> 29
*3(34, 35) -> 30
*3(34, 35) -> 31
+2(28, 29) -> 29
+2(28, 29) -> 30
+2(28, 29) -> 31
f2(28) -> 30
f2(29) -> 31
*2(30, 31) -> 29
*2(30, 31) -> 30
*2(30, 31) -> 31
f3(28) -> 37
f3(29) -> 38
*3(37, 38) -> 29
*3(37, 38) -> 30
*3(37, 38) -> 31
f2(28) -> 29
s3(33) -> 39
f3(28) -> 40
+3(39, 40) -> 29
+3(39, 40) -> 30
+3(39, 40) -> 31
04() -> 41
s4(41) -> 35
s3(33) -> 38
*3(34, 35) -> 38
+3(39, 40) -> 38
*3(37, 38) -> 38
f3(39) -> 37
f3(40) -> 38
f4(39) -> 42
f4(40) -> 43
*4(42, 43) -> 38

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

(8)
BOUNDS(1, n^1)

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(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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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

The rewrite sequence

f(+(x, y)) ->^+ *(f(x), f(y))

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

The pumping substitution is [x / +(x, y)].

The result substitution is [ ].




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(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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(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:

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST