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), 0 ms)
↳2 CpxTRS
↳3 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxTRS
↳5 TrsToWeightedTrsProof (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 IntTrsBoundProof (UPPER BOUND(ID), 223 ms)
↳20 CpxRNTS
↳21 IntTrsBoundProof (UPPER BOUND(ID), 9 ms)
↳22 CpxRNTS
↳23 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 58 ms)
↳26 CpxRNTS
↳27 IntTrsBoundProof (UPPER BOUND(ID), 28 ms)
↳28 CpxRNTS
↳29 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 106 ms)
↳32 CpxRNTS
↳33 IntTrsBoundProof (UPPER BOUND(ID), 97 ms)
↳34 CpxRNTS
↳35 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳36 CpxRNTS
↳37 IntTrsBoundProof (UPPER BOUND(ID), 192 ms)
↳38 CpxRNTS
↳39 IntTrsBoundProof (UPPER BOUND(ID), 49 ms)
↳40 CpxRNTS
↳41 FinalProof (⇔, 0 ms)
↳42 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:

rec(rec(x)) → sent(rec(x))
rec(sent(x)) → sent(rec(x))
rec(no(x)) → sent(rec(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
sent(up(x)) → up(sent(x))
no(up(x)) → up(no(x))
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))
check(up(x)) → up(check(x))
check(sent(x)) → sent(check(x))
check(rec(x)) → rec(check(x))
check(no(x)) → no(check(x))
check(no(x)) → no(x)

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:
top(rec(up(x))) → top(check(rec(x)))
top(sent(up(x))) → top(check(rec(x)))
top(no(up(x))) → top(check(rec(x)))

(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:

check(no(x)) → no(x)
rec(sent(x)) → sent(rec(x))
check(sent(x)) → sent(check(x))
check(up(x)) → up(check(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
rec(rec(x)) → sent(rec(x))
check(no(x)) → no(check(x))
no(up(x)) → up(no(x))
rec(no(x)) → sent(rec(x))
sent(up(x)) → up(sent(x))
check(rec(x)) → rec(check(x))

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
check(no(x)) → no(x)
rec(sent(x)) → sent(rec(x))
check(sent(x)) → sent(check(x))
rec(rec(x)) → sent(rec(x))
check(no(x)) → no(check(x))
rec(no(x)) → sent(rec(x))
check(rec(x)) → rec(check(x))

(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:

check(up(x)) → up(check(x))
rec(bot) → up(sent(bot))
rec(up(x)) → up(rec(x))
no(up(x)) → up(no(x))
sent(up(x)) → up(sent(x))

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(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:

check(up(x)) → up(check(x)) [1]
rec(bot) → up(sent(bot)) [1]
rec(up(x)) → up(rec(x)) [1]
no(up(x)) → up(no(x)) [1]
sent(up(x)) → up(sent(x)) [1]

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:

check(up(x)) → up(check(x)) [1]
rec(bot) → up(sent(bot)) [1]
rec(up(x)) → up(rec(x)) [1]
no(up(x)) → up(no(x)) [1]
sent(up(x)) → up(sent(x)) [1]

The TRS has the following type information:
check :: up:bot → up:bot
up :: up:bot → up:bot
rec :: up:bot → up:bot
bot :: up:bot
sent :: up:bot → up:bot
no :: up:bot → up:bot

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:


check
rec
no
sent

(c) The following functions are completely defined:
none

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:

check(up(x)) → up(check(x)) [1]
rec(bot) → up(sent(bot)) [1]
rec(up(x)) → up(rec(x)) [1]
no(up(x)) → up(no(x)) [1]
sent(up(x)) → up(sent(x)) [1]

The TRS has the following type information:
check :: up:bot → up:bot
up :: up:bot → up:bot
rec :: up:bot → up:bot
bot :: up:bot
sent :: up:bot → up:bot
no :: up:bot → up:bot

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:

check(up(x)) → up(check(x)) [1]
rec(bot) → up(sent(bot)) [1]
rec(up(x)) → up(rec(x)) [1]
no(up(x)) → up(no(x)) [1]
sent(up(x)) → up(sent(x)) [1]

The TRS has the following type information:
check :: up:bot → up:bot
up :: up:bot → up:bot
rec :: up:bot → up:bot
bot :: up:bot
sent :: up:bot → up:bot
no :: up:bot → up:bot

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:

bot => 0

(14) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ 1 }→ 1 + check(x) :|: x >= 0, z = 1 + x
no(z) -{ 1 }→ 1 + no(x) :|: x >= 0, z = 1 + x
rec(z) -{ 1 }→ 1 + sent(0) :|: z = 0
rec(z) -{ 1 }→ 1 + rec(x) :|: x >= 0, z = 1 + x
sent(z) -{ 1 }→ 1 + sent(x) :|: x >= 0, z = 1 + x

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ 1 }→ 1 + check(z - 1) :|: z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + sent(0) :|: z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ 1 }→ 1 + sent(z - 1) :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ sent }
{ check }
{ no }
{ rec }

(18) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ 1 }→ 1 + check(z - 1) :|: z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + sent(0) :|: z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ 1 }→ 1 + sent(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {sent}, {check}, {no}, {rec}

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(20) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ 1 }→ 1 + check(z - 1) :|: z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + sent(0) :|: z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ 1 }→ 1 + sent(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {sent}, {check}, {no}, {rec}
Previous analysis results are:
sent: runtime: ?, size: O(1) [0]

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


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

(22) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ 1 }→ 1 + check(z - 1) :|: z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + sent(0) :|: z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ 1 }→ 1 + sent(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {check}, {no}, {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ 1 }→ 1 + check(z - 1) :|: z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {check}, {no}, {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(26) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ 1 }→ 1 + check(z - 1) :|: z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {check}, {no}, {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: ?, size: O(1) [0]

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


Computed RUNTIME bound using CoFloCo for: check
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:

check(z) -{ 1 }→ 1 + check(z - 1) :|: z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {no}, {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: O(n1) [z], size: O(1) [0]

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ z }→ 1 + s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {no}, {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: O(n1) [z], size: O(1) [0]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: no
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(32) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ z }→ 1 + s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {no}, {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: O(n1) [z], size: O(1) [0]
no: runtime: ?, size: O(1) [0]

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


Computed RUNTIME bound using CoFloCo for: no
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:

check(z) -{ z }→ 1 + s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
no(z) -{ 1 }→ 1 + no(z - 1) :|: z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: O(n1) [z], size: O(1) [0]
no: runtime: O(n1) [z], size: O(1) [0]

(35) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ z }→ 1 + s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
no(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: O(n1) [z], size: O(1) [0]
no: runtime: O(n1) [z], size: O(1) [0]

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(38) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ z }→ 1 + s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
no(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed: {rec}
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: O(n1) [z], size: O(1) [0]
no: runtime: O(n1) [z], size: O(1) [0]
rec: runtime: ?, size: O(n1) [1 + z]

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


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

(40) Obligation:

Complexity RNTS consisting of the following rules:

check(z) -{ z }→ 1 + s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
no(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 0, z - 1 >= 0
rec(z) -{ 1 }→ 1 + s :|: s >= 0, s <= 0, z = 0
rec(z) -{ 1 }→ 1 + rec(z - 1) :|: z - 1 >= 0
sent(z) -{ z }→ 1 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:
sent: runtime: O(n1) [z], size: O(1) [0]
check: runtime: O(n1) [z], size: O(1) [0]
no: runtime: O(n1) [z], size: O(1) [0]
rec: runtime: O(n1) [1 + z], size: O(n1) [1 + z]

(41) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(42) BOUNDS(1, n^1)