/export/starexec/sandbox2/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 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), 236 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
    (24) CpxRNTS
    (25) FinalProof [FINISHED, 0 ms]
    (26) BOUNDS(1, n^1)
(27) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(28) CpxTRS
    (29) SlicingProof [LOWER BOUND(ID), 0 ms]
    (30) CpxTRS
    (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) typed CpxTrs
    (33) OrderProof [LOWER BOUND(ID), 0 ms]
    (34) typed CpxTrs
    (35) RewriteLemmaProof [LOWER BOUND(ID), 463 ms]
    (36) proven lower bound
    (37) LowerBoundPropagationProof [FINISHED, 0 ms]
    (38) BOUNDS(n^1, INF)


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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:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))

S is empty.
Rewrite Strategy: FULL
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(1) 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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(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:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))

S is empty.
Rewrite Strategy: INNERMOST
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(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:

   *(x, +(y, z)) -> +(*(x, y), *(x, z)) [1]

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

* => times

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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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]

The TRS has the following type information:
times :: a -> + -> +
+ :: + -> + -> +

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:

   times_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]

The TRS has the following type information:
times :: a -> + -> +
+ :: + -> + -> +
const :: +
const1 :: a

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

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]

The TRS has the following type information:
times :: a -> + -> +
+ :: + -> + -> +
const :: +
const1 :: a

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

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

   const => 0
   const1 => 0

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

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


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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:

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


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

   { times }

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

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

Function symbols to be analyzed: {times}

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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:

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

Function symbols to be analyzed: {times}

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(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

Function symbols to be analyzed: {times}
Previous analysis results are:
  times: runtime: ?, size: O(1) [0]

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(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

Function symbols to be analyzed: 
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(1) [0]

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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) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(28)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   *'(x, +'(y, z)) -> +'(*'(x, y), *'(x, z))

S is empty.
Rewrite Strategy: FULL
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(29) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
*'/0

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(30)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   *'(+'(y, z)) -> +'(*'(y), *'(z))

S is empty.
Rewrite Strategy: FULL
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(31) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(32)
Obligation:
TRS:
Rules:
*'(+'(y, z)) -> +'(*'(y), *'(z))

Types:
*' :: +' -> +'
+' :: +' -> +' -> +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat -> +'

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(33) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
*'
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(34)
Obligation:
TRS:
Rules:
*'(+'(y, z)) -> +'(*'(y), *'(z))

Types:
*' :: +' -> +'
+' :: +' -> +' -> +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat -> +'


Generator Equations:
gen_+'2_0(0) <=> hole_+'1_0
gen_+'2_0(+(x, 1)) <=> +'(hole_+'1_0, gen_+'2_0(x))


The following defined symbols remain to be analysed:
*'
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(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
*'(gen_+'2_0(+(1, 0)))

Induction Step:
*'(gen_+'2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
+'(*'(hole_+'1_0), *'(gen_+'2_0(+(1, n4_0)))) ->_IH
+'(*'(hole_+'1_0), *3_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(36)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
*'(+'(y, z)) -> +'(*'(y), *'(z))

Types:
*' :: +' -> +'
+' :: +' -> +' -> +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat -> +'


Generator Equations:
gen_+'2_0(0) <=> hole_+'1_0
gen_+'2_0(+(x, 1)) <=> +'(hole_+'1_0, gen_+'2_0(x))


The following defined symbols remain to be analysed:
*'
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(37) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(38)
BOUNDS(n^1, INF)