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(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(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(g(X), Y) → f(X, n__f(n__g(X), activate(Y))) [1]
f(X1, X2) → n__f(X1, X2) [1]
g(X) → n__g(X) [1]
activate(n__f(X1, X2)) → f(activate(X1), X2) [1]
activate(n__g(X)) → g(activate(X)) [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

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

f(g(X), Y) → f(X, n__f(n__g(X), activate(Y))) [1]

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

g(X) → n__g(X) [1]

(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(X1, X2) → n__f(X1, X2) [1]
g(X) → n__g(X) [1]
activate(n__f(X1, X2)) → f(activate(X1), X2) [1]
activate(n__g(X)) → g(activate(X)) [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

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

f(X1, X2) → n__f(X1, X2) [1]
g(X) → n__g(X) [1]
activate(n__f(X1, X2)) → f(activate(X1), X2) [1]
activate(n__g(X)) → g(activate(X)) [1]
activate(X) → X [1]

The TRS has the following type information:

f :: n__f:n__g → a → n__f:n__g n__f :: n__f:n__g → a → n__f:n__g g :: n__f:n__g → n__f:n__g n__g :: n__f:n__g → n__f:n__g activate :: n__f:n__g → n__f:n__g

Rewrite Strategy: INNERMOST

(7) 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
g
f

Due to the following rules being added:
none

And the following fresh constants:

const, const1

(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(X1, X2) → n__f(X1, X2) [1] g(X) → n__g(X) [1] activate(n__f(X1, X2)) → f(activate(X1), X2) [1] activate(n__g(X)) → g(activate(X)) [1] activate(X) → X [1] The TRS has the following type information: f :: n__f:n__g → a → n__f:n__g n__f :: n__f:n__g → a → n__f:n__g g :: n__f:n__g → n__f:n__g n__g :: n__f:n__g → n__f:n__g activate :: n__f:n__g → n__f:n__g const :: n__f:n__g const1 :: a Rewrite Strategy: INNERMOST

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

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(X1, X2) → n__f(X1, X2) [1] g(X) → n__g(X) [1] activate(n__f(n__f(X1', X2'), X2)) → f(f(activate(X1'), X2'), X2) [2] activate(n__f(n__g(X'), X2)) → f(g(activate(X')), X2) [2] activate(n__f(X1, X2)) → f(X1, X2) [2] activate(n__g(n__f(X1'', X2''))) → g(f(activate(X1''), X2'')) [2] activate(n__g(n__g(X''))) → g(g(activate(X''))) [2] activate(n__g(X)) → g(X) [2] activate(X) → X [1] The TRS has the following type information: f :: n__f:n__g → a → n__f:n__g n__f :: n__f:n__g → a → n__f:n__g g :: n__f:n__g → n__f:n__g n__g :: n__f:n__g → n__f:n__g activate :: n__f:n__g → n__f:n__g const :: n__f:n__g const1 :: a Rewrite Strategy: INNERMOST

(11) 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
const1 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

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

(13) InliningProof (UPPER BOUND(ID) transformation)

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

Found the following analysis order by SCC decomposition:

{ g } { f } { activate }

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(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:

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

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

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

(25) 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: 1 + z + z'

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: ?, size: O(n1) [1 + z + z']

(27) 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

(28) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [1 + z + z']

(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:

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

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [1 + z + z']

(31) 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: z

(32) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [1 + z + z'] activate: runtime: ?, size: O(n1) [z]

(33) 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: 7 + 16·z

(34) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [1], size: O(n1) [1 + z + z'] activate: runtime: O(n1) [7 + 16·z], size: O(n1) [z]

(35) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(36) BOUNDS(1, n^1)