Runtime Complexity (innermost) proof of /tmp/tmptbWVN6/Ex7_BLR02

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

0 CpxTRS

↳1 DependencyGraphProof (BOTH BOUNDS(ID, ID), 9 ms)

↳2 CpxTRS

↳3 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CpxWeightedTrs

↳5 InnermostUnusableRulesProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CpxWeightedTrs

↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CpxTypedWeightedTrs

↳9 CompletionProof (UPPER BOUND(ID), 4 ms)

↳10 CpxTypedWeightedCompleteTrs

↳11 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CpxTypedWeightedCompleteTrs

↳13 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 7 ms)

↳14 CpxRNTS

↳15 InliningProof (UPPER BOUND(ID), 170 ms)

↳16 CpxRNTS

↳17 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 CpxRNTS

↳19 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CpxRNTS

↳21 IntTrsBoundProof (UPPER BOUND(ID), 258 ms)

↳22 CpxRNTS

↳23 IntTrsBoundProof (UPPER BOUND(ID), 66 ms)

↳24 CpxRNTS

↳25 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳26 CpxRNTS

↳27 IntTrsBoundProof (UPPER BOUND(ID), 168 ms)

↳28 CpxRNTS

↳29 IntTrsBoundProof (UPPER BOUND(ID), 58 ms)

↳30 CpxRNTS

↳31 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳32 CpxRNTS

↳33 IntTrsBoundProof (UPPER BOUND(ID), 206 ms)

↳34 CpxRNTS

↳35 IntTrsBoundProof (UPPER BOUND(ID), 5 ms)

↳36 CpxRNTS

↳37 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳38 CpxRNTS

↳39 IntTrsBoundProof (UPPER BOUND(ID), 227 ms)

↳40 CpxRNTS

↳41 IntTrsBoundProof (UPPER BOUND(ID), 28 ms)

↳42 CpxRNTS

↳43 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳44 CpxRNTS

↳45 IntTrsBoundProof (UPPER BOUND(ID), 96 ms)

↳46 CpxRNTS

↳47 IntTrsBoundProof (UPPER BOUND(ID), 97 ms)

↳48 CpxRNTS

↳49 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳50 CpxRNTS

↳51 IntTrsBoundProof (UPPER BOUND(ID), 8699 ms)

↳52 CpxRNTS

↳53 IntTrsBoundProof (UPPER BOUND(ID), 1469 ms)

↳54 CpxRNTS

↳55 ResultPropagationProof (UPPER BOUND(ID), 3 ms)

↳56 CpxRNTS

↳57 IntTrsBoundProof (UPPER BOUND(ID), 1205 ms)

↳58 CpxRNTS

↳59 IntTrsBoundProof (UPPER BOUND(ID), 33 ms)

↳60 CpxRNTS

↳61 FinalProof (⇔, 0 ms)

↳62 BOUNDS(1, n^1)

(0) 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:

Rewrite Strategy: INNERMOST

(1) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)

The following rules are not reachable from basic terms in the dependency graph and can be removed:
sel(s(N), cons(X, XS)) → sel(N, activate(XS))

(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:

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(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:

Rewrite Strategy: INNERMOST

(5) InnermostUnusableRulesProof (BOTH BOUNDS(ID, ID) transformation)

Removed the following rules with non-basic left-hand side, as they cannot be used in innermost rewriting:

Due to the following rules that have to be used instead:

(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:

Rewrite Strategy: INNERMOST

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(8) Obligation:

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

The TRS has the following type information:

sel :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
0 :: 0:cons:n__s:n__take:n__from:nil
cons :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
head :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
activate :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
n__s :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
s :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
take :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
n__take :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
n__from :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
from :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil
nil :: 0:cons:n__s:n__take:n__from:nil
2nd :: 0:cons:n__s:n__take:n__from:nil → 0:cons:n__s:n__take:n__from:nil

Rewrite Strategy: INNERMOST

(9) CompletionProof (UPPER BOUND(ID) transformation)

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:

Due to the following rules being added:
none

And the following fresh constants: none

(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:

The TRS has the following type information:

Rewrite Strategy: INNERMOST

(11) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

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:

The TRS has the following type information:

Rewrite Strategy: INNERMOST

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

(15) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

(16) Obligation:

Complexity RNTS consisting of the following rules:

(17) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(18) Obligation:

Complexity RNTS consisting of the following rules:

(19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ from }
{ sel }
{ head }
{ take }
{ s }
{ activate }
{ 2nd }

(20) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {from}, {sel}, {head}, {take}, {s}, {activate}, {2nd}

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + 2·z

(22) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {from}, {sel}, {head}, {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: ?, size: O(n¹) [3 + 2·z]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {sel}, {head}, {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(26) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {sel}, {head}, {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: sel
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z'

(28) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {sel}, {head}, {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: ?, size: O(n¹) [z']

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {head}, {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(32) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {head}, {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(34) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {head}, {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']
head: runtime: ?, size: O(n¹) [z]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']
head: runtime: O(1) [1], size: O(n¹) [z]

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(38) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']
head: runtime: O(1) [1], size: O(n¹) [z]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: take
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z + z'

(40) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {take}, {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']
head: runtime: O(1) [1], size: O(n¹) [z]
take: runtime: ?, size: O(n¹) [1 + z + z']

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(42) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']
head: runtime: O(1) [1], size: O(n¹) [z]
take: runtime: O(1) [1], size: O(n¹) [1 + z + z']

(43) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(44) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {s}, {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']
head: runtime: O(1) [1], size: O(n¹) [z]
take: runtime: O(1) [1], size: O(n¹) [1 + z + z']

(45) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: s
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

(46) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {s}, {activate}, {2nd}
Previous analysis results are:

(47) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(48) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {activate}, {2nd}
Previous analysis results are:

(49) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(50) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {activate}, {2nd}
Previous analysis results are:

(51) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(52) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {activate}, {2nd}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n¹) [3 + 2·z]
sel: runtime: O(1) [1], size: O(n¹) [z']
head: runtime: O(1) [1], size: O(n¹) [z]
take: runtime: O(1) [1], size: O(n¹) [1 + z + z']
s: runtime: O(1) [1], size: O(n¹) [1 + z]
activate: runtime: ?, size: EXP

(53) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 108 + 216·z

(54) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {2nd}
Previous analysis results are:

(55) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(56) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {2nd}
Previous analysis results are:

(57) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: 2nd
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(58) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed: {2nd}
Previous analysis results are:

(59) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: 2nd
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 447 + 864·z

(60) Obligation:

Complexity RNTS consisting of the following rules:

Function symbols to be analyzed:
Previous analysis results are:

(61) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(0) Obligation:

(1) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)

(2) Obligation:

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

(4) Obligation:

(5) InnermostUnusableRulesProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CompletionProof (UPPER BOUND(ID) transformation)

(10) Obligation:

(11) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(12) Obligation:

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

(14) Obligation:

(15) InliningProof (UPPER BOUND(ID) transformation)

(16) Obligation:

(17) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

(18) Obligation:

(19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

(20) Obligation:

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(22) Obligation:

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(24) Obligation:

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

(26) Obligation:

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(28) Obligation:

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(30) Obligation:

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

(32) Obligation:

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(34) Obligation:

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(36) Obligation:

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

(38) Obligation:

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(40) Obligation:

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(42) Obligation:

(43) ResultPropagationProof (UPPER BOUND(ID) transformation)

(44) Obligation:

(45) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(46) Obligation:

(47) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(48) Obligation:

(49) ResultPropagationProof (UPPER BOUND(ID) transformation)

(50) Obligation:

(51) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(52) Obligation:

(53) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(54) Obligation:

(55) ResultPropagationProof (UPPER BOUND(ID) transformation)

(56) Obligation:

(57) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(58) Obligation:

(59) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(60) Obligation:

(61) FinalProof (EQUIVALENT transformation)

(62) BOUNDS(1, n^1)