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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 402 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 220 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
    (30) CpxRNTS
    (31) FinalProof [FINISHED, 0 ms]
    (32) BOUNDS(1, n^1)
(33) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(34) TRS for Loop Detection
    (35) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (36) BEST
        (37) proven lower bound
            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
            (39) BOUNDS(n^1, INF)
        (40) TRS for Loop Detection


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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

(2)
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:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

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

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]
   +(s(x), y) -> s(+(x, y)) [1]
   double(x) -> +(x, x) [1]

Rewrite Strategy: INNERMOST
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(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

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


The TRS R consists of the following rules:

   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   double(x) -> plus(x, x) [1]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   double(x) -> plus(x, x) [1]

The TRS has the following type information:
double :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   double_1
   plus_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   double(x) -> plus(x, x) [1]

The TRS has the following type information:
double :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
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(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   double(x) -> plus(x, x) [1]

The TRS has the following type information:
double :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
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(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0

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(14)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> plus(x, x) :|: x >= 0, z = x
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y


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(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0


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(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { plus }
   { double }

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(18)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus}, {double}

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(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus}, {double}

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(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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(22)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus}, {double}
Previous analysis results are:
  plus: runtime: ?, size: O(n^1) [z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z'

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> plus(z, z) :|: z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {double}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']

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(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 2 + 2*z }-> s'' :|: s'' >= 0, s'' <= z + z, z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {double}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: double
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2*z

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 2 + 2*z }-> s'' :|: s'' >= 0, s'' <= z + z, z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {double}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
  double: runtime: ?, size: O(n^1) [2*z]

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(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: double
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 2*z

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(30)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 2 + 2*z }-> s'' :|: s'' >= 0, s'' <= z + z, z >= 0
   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z + z'], size: O(n^1) [z + z']
  double: runtime: O(n^1) [2 + 2*z], size: O(n^1) [2*z]

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(31) FinalProof (FINISHED)
Computed overall runtime complexity
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(32)
BOUNDS(1, n^1)

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

(34)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

S is empty.
Rewrite Strategy: FULL
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(35) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

+(x, s(y)) ->^+ s(+(x, y))

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

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

The result substitution is [ ].




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(36)
Complex Obligation (BEST)

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(37)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

S is empty.
Rewrite Strategy: FULL
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(38) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(39)
BOUNDS(n^1, INF)

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(40)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

S is empty.
Rewrite Strategy: FULL