/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 279 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
    (22) CpxRNTS
    (23) FinalProof [FINISHED, 0 ms]
    (24) BOUNDS(1, n^1)
(25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CpxTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 0 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 437 ms]
    (32) proven lower bound
    (33) LowerBoundPropagationProof [FINISHED, 0 ms]
    (34) 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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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:

   f(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]

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

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]

The TRS has the following type information:
f :: g:h -> g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h

Rewrite Strategy: INNERMOST
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(7) 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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(8)
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(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]

The TRS has the following type information:
f :: g:h -> g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
const :: g:h

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

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

(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(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]

The TRS has the following type information:
f :: g:h -> g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
const :: g:h

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

(11) 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

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(12)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

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

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


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

   { f }

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

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

Function symbols to be analyzed: {f}

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(17) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(18)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {f}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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(23) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(24)
BOUNDS(1, n^1)

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

(26)
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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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

(27) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(28)
Obligation:
TRS:
Rules:
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)

Types:
f :: g:h -> g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
hole_g:h1_0 :: g:h
gen_g:h2_0 :: Nat -> g:h

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(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f
----------------------------------------

(30)
Obligation:
TRS:
Rules:
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)

Types:
f :: g:h -> g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
hole_g:h1_0 :: g:h
gen_g:h2_0 :: Nat -> g:h


Generator Equations:
gen_g:h2_0(0) <=> hole_g:h1_0
gen_g:h2_0(+(x, 1)) <=> h(gen_g:h2_0(x))


The following defined symbols remain to be analysed:
f
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(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, 0)))

Induction Step:
f(gen_g:h2_0(a), gen_g:h2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
f(h(gen_g:h2_0(a)), gen_g:h2_0(+(1, n4_0))) ->_IH
*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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(32)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)

Types:
f :: g:h -> g:h -> g:h
g :: g:h -> g:h
h :: g:h -> g:h
hole_g:h1_0 :: g:h
gen_g:h2_0 :: Nat -> g:h


Generator Equations:
gen_g:h2_0(0) <=> hole_g:h1_0
gen_g:h2_0(+(x, 1)) <=> h(gen_g:h2_0(x))


The following defined symbols remain to be analysed:
f
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(33) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(34)
BOUNDS(n^1, INF)