The Runtime Complexity (innermost) of the given CpxTRS could be proven to be 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:

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) 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(n__h(n__f(X))) [1]
h(X) → n__h(X) [1]
f(X) → n__f(X) [1]
activate(n__h(X)) → h(activate(X)) [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

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

f(X) → g(n__h(n__f(X))) [1]
h(X) → n__h(X) [1]
f(X) → n__f(X) [1]
activate(n__h(X)) → h(activate(X)) [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(X) → X [1]

The TRS has the following type information:

f :: n__f:n__h:g → n__f:n__h:g g :: n__f:n__h:g → n__f:n__h:g n__h :: n__f:n__h:g → n__f:n__h:g n__f :: n__f:n__h:g → n__f:n__h:g h :: n__f:n__h:g → n__f:n__h:g activate :: n__f:n__h:g → n__f:n__h:g

Rewrite Strategy: INNERMOST

(5) 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:
none

(c) The following functions are completely defined:

activate
h
f

Due to the following rules being added:
none

And the following fresh constants:

const

(6) 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(n__h(n__f(X))) [1] h(X) → n__h(X) [1] f(X) → n__f(X) [1] activate(n__h(X)) → h(activate(X)) [1] activate(n__f(X)) → f(activate(X)) [1] activate(X) → X [1] The TRS has the following type information: f :: n__f:n__h:g → n__f:n__h:g g :: n__f:n__h:g → n__f:n__h:g n__h :: n__f:n__h:g → n__f:n__h:g n__f :: n__f:n__h:g → n__f:n__h:g h :: n__f:n__h:g → n__f:n__h:g activate :: n__f:n__h:g → n__f:n__h:g const :: n__f:n__h:g Rewrite Strategy: INNERMOST

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

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(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(n__h(n__f(X))) [1] h(X) → n__h(X) [1] f(X) → n__f(X) [1] activate(n__h(n__h(X'))) → h(h(activate(X'))) [2] activate(n__h(n__f(X''))) → h(f(activate(X''))) [2] activate(n__h(X)) → h(X) [2] activate(n__f(n__h(X1))) → f(h(activate(X1))) [2] activate(n__f(n__f(X2))) → f(f(activate(X2))) [2] activate(n__f(X)) → f(X) [2] activate(X) → X [1] The TRS has the following type information: f :: n__f:n__h:g → n__f:n__h:g g :: n__f:n__h:g → n__f:n__h:g n__h :: n__f:n__h:g → n__f:n__h:g n__f :: n__f:n__h:g → n__f:n__h:g h :: n__f:n__h:g → n__f:n__h:g activate :: n__f:n__h:g → n__f:n__h:g const :: n__f:n__h:g Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ h(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ h(h(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 2 }→ h(f(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ f(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ f(h(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 2 }→ f(f(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0
f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
f(z) -{ 1 }→ 1 + (1 + (1 + X)) :|: X >= 0, z = X
h(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(11) InliningProof (UPPER BOUND(ID) transformation)

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

f(z) -{ 1 }→ 1 + (1 + (1 + X)) :|: X >= 0, z = X
f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
h(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(12) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ h(h(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 2 }→ h(f(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ f(h(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 2 }→ f(f(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
f(z) -{ 1 }→ 1 + (1 + (1 + X)) :|: X >= 0, z = X
h(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ h } { f } { activate }

(16) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {h}, {f}, {activate}

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {h}, {f}, {activate}
Previous analysis results are:

h: runtime: ?, size: O(n1) [1 + z]

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

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

h: runtime: O(1) [1], size: O(n1) [1 + z]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

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

h: runtime: O(1) [1], size: O(n1) [1 + z]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

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

h: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: ?, size: O(n1) [3 + z]

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

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

h: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [3 + z]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

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

h: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [3 + z]

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

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

h: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [3 + z] activate: runtime: ?, size: O(n1) [3·z]

(31) 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: 4 + 4·z

(32) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ h(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ h(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(h(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + (1 + (1 + z)) :|: z >= 0
h(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:

h: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [3 + z] activate: runtime: O(n1) [4 + 4·z], size: O(n1) [3·z]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^1)