/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [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), 952 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 212 ms]
    (24) CpxRNTS
    (25) FinalProof [FINISHED, 0 ms]
    (26) BOUNDS(1, n^1)
(27) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(28) TRS for Loop Detection
    (29) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) 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:

   f(0, y) -> y
   f(x, 0) -> x
   f(i(x), y) -> i(x)
   f(f(x, y), z) -> f(x, f(y, z))
   f(g(x, y), z) -> g(f(x, z), f(y, z))
   f(1, g(x, y)) -> x
   f(2, g(x, y)) -> y

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(f(x, y), z) -> f(x, f(y, z))

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(2)
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:

   f(0, y) -> y
   f(x, 0) -> x
   f(i(x), y) -> i(x)
   f(g(x, y), z) -> g(f(x, z), f(y, z))
   f(1, g(x, y)) -> x
   f(2, g(x, y)) -> y

S is empty.
Rewrite Strategy: FULL
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(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
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(4)
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, y) -> y
   f(x, 0) -> x
   f(i(x), y) -> i(x)
   f(g(x, y), z) -> g(f(x, z), f(y, z))
   f(1, g(x, y)) -> x
   f(2, g(x, y)) -> y

S is empty.
Rewrite Strategy: INNERMOST
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(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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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:

   f(0, y) -> y [1]
   f(x, 0) -> x [1]
   f(i(x), y) -> i(x) [1]
   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
   f(1, g(x, y)) -> x [1]
   f(2, g(x, y)) -> y [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:

   f(0, y) -> y [1]
   f(x, 0) -> x [1]
   f(i(x), y) -> i(x) [1]
   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
   f(1, g(x, y)) -> x [1]
   f(2, g(x, y)) -> y [1]

The TRS has the following type information:
f :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
0 :: 0:i:g:1:2
i :: a -> 0:i:g:1:2
g :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
1 :: 0:i:g:1:2
2 :: 0:i:g:1:2

Rewrite Strategy: INNERMOST
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(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:

   f_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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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:

   f(0, y) -> y [1]
   f(x, 0) -> x [1]
   f(i(x), y) -> i(x) [1]
   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
   f(1, g(x, y)) -> x [1]
   f(2, g(x, y)) -> y [1]

The TRS has the following type information:
f :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
0 :: 0:i:g:1:2
i :: a -> 0:i:g:1:2
g :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
1 :: 0:i:g:1:2
2 :: 0:i:g:1:2
const :: a

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:

   f(0, y) -> y [1]
   f(x, 0) -> x [1]
   f(i(x), y) -> i(x) [1]
   f(g(x, y), z) -> g(f(x, z), f(y, z)) [1]
   f(1, g(x, y)) -> x [1]
   f(2, g(x, y)) -> y [1]

The TRS has the following type information:
f :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
0 :: 0:i:g:1:2
i :: a -> 0:i:g:1:2
g :: 0:i:g:1:2 -> 0:i:g:1:2 -> 0:i:g:1:2
1 :: 0:i:g:1:2
2 :: 0:i:g:1:2
const :: a

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
   1 => 1
   2 => 2
   const => 0

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

   f(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
   f(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
   f(z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   f(z', z'') -{ 1 }-> 1 + f(x, z) + f(y, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0


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(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(16)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0


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

   { f }

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

   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}

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(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(20)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}

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(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 8*z' + 4*z'*z'' + 2*z''

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

   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  f: runtime: ?, size: O(n^2) [2 + 8*z' + 4*z'*z'' + 2*z'']

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(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z', z'') -{ 1 }-> x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
   f(z', z'') -{ 1 }-> y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
   f(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   f(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   f(z', z'') -{ 1 }-> 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
   f(z', z'') -{ 1 }-> 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  f: runtime: O(n^1) [1 + 2*z'], size: O(n^2) [2 + 8*z' + 4*z'*z'' + 2*z'']

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(25) FinalProof (FINISHED)
Computed overall runtime complexity
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(26)
BOUNDS(1, n^1)

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(27) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(28)
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:

   f(0, y) -> y
   f(x, 0) -> x
   f(i(x), y) -> i(x)
   f(f(x, y), z) -> f(x, f(y, z))
   f(g(x, y), z) -> g(f(x, z), f(y, z))
   f(1, g(x, y)) -> x
   f(2, g(x, y)) -> y

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

The rewrite sequence

f(g(x, y), z) ->^+ g(f(x, z), f(y, z))

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

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

The result substitution is [ ].




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(30)
Complex Obligation (BEST)

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(31)
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:

   f(0, y) -> y
   f(x, 0) -> x
   f(i(x), y) -> i(x)
   f(f(x, y), z) -> f(x, f(y, z))
   f(g(x, y), z) -> g(f(x, z), f(y, z))
   f(1, g(x, y)) -> x
   f(2, g(x, y)) -> y

S is empty.
Rewrite Strategy: FULL
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(32) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(33)
BOUNDS(n^1, INF)

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(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:

   f(0, y) -> y
   f(x, 0) -> x
   f(i(x), y) -> i(x)
   f(f(x, y), z) -> f(x, f(y, z))
   f(g(x, y), z) -> g(f(x, z), f(y, z))
   f(1, g(x, y)) -> x
   f(2, g(x, y)) -> y

S is empty.
Rewrite Strategy: FULL