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

(0) Obligation:

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


The TRS R consists of the following rules:

f(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(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, 1).


The TRS R consists of the following rules:

f(n__f(n__a)) → f(n__g(f(n__a))) [1]
f(X) → n__f(X) [1]
a → n__a [1]
g(X) → n__g(X) [1]
activate(n__f(X)) → f(X) [1]
activate(n__a) → a [1]
activate(n__g(X)) → g(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(n__f(n__a)) → f(n__g(f(n__a))) [1]
f(X) → n__f(X) [1]
a → n__a [1]
g(X) → n__g(X) [1]
activate(n__f(X)) → f(X) [1]
activate(n__a) → a [1]
activate(n__g(X)) → g(X) [1]
activate(X) → X [1]

The TRS has the following type information:

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

a
g
activate

(c) The following functions are completely defined:

f

Due to the following rules being added:
none

And the following fresh constants: none

(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(n__f(n__a)) → f(n__g(f(n__a))) [1] f(X) → n__f(X) [1] a → n__a [1] g(X) → n__g(X) [1] activate(n__f(X)) → f(X) [1] activate(n__a) → a [1] activate(n__g(X)) → g(X) [1] activate(X) → X [1] The TRS has the following type information: f :: n__a:n__f:n__g → n__a:n__f:n__g n__f :: n__a:n__f:n__g → n__a:n__f:n__g n__a :: n__a:n__f:n__g n__g :: n__a:n__f:n__g → n__a:n__f:n__g a :: n__a:n__f:n__g g :: n__a:n__f:n__g → n__a:n__f:n__g activate :: n__a:n__f:n__g → n__a:n__f:n__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(n__f(n__a)) → f(n__g(n__f(n__a))) [2] f(X) → n__f(X) [1] a → n__a [1] g(X) → n__g(X) [1] activate(n__f(X)) → f(X) [1] activate(n__a) → a [1] activate(n__g(X)) → g(X) [1] activate(X) → X [1] The TRS has the following type information: f :: n__a:n__f:n__g → n__a:n__f:n__g n__f :: n__a:n__f:n__g → n__a:n__f:n__g n__a :: n__a:n__f:n__g n__g :: n__a:n__f:n__g → n__a:n__f:n__g a :: n__a:n__f:n__g g :: n__a:n__f:n__g → n__a:n__f:n__g activate :: n__a:n__f:n__g → n__a:n__f:n__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:

n__a => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

Found the following analysis order by SCC decomposition:

{ g } { f } { a } { activate }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(19) 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

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: 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:

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

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

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: ?, size: O(n1) [2 + 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: 3

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [3], size: O(n1) [2 + 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:

a -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
g(z) -{ 1 }→ 1 + z :|: z >= 0

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [3], size: O(n1) [2 + z]

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
g(z) -{ 1 }→ 1 + z :|: z >= 0

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

g: runtime: O(1) [1], size: O(n1) [1 + z] f: runtime: O(1) [3], size: O(n1) [2 + z] a: runtime: ?, size: O(1) [0]

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

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
f(z) -{ 1 }→ 1 + z :|: 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) [3], size: O(n1) [2 + z] a: runtime: O(1) [1], size: O(1) [0]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
f(z) -{ 1 }→ 1 + z :|: 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) [3], size: O(n1) [2 + z] a: runtime: O(1) [1], size: O(1) [0]

(35) 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: 1 + z

(36) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
f(z) -{ 1 }→ 1 + z :|: 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) [3], size: O(n1) [2 + z] a: runtime: O(1) [1], size: O(1) [0] activate: runtime: ?, size: O(n1) [1 + z]

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

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 9

(38) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 2, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 5 }→ s :|: s >= 0, s <= 1 * (1 + (1 + 0)) + 2, z = 1 + 0
f(z) -{ 1 }→ 1 + z :|: 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) [3], size: O(n1) [2 + z] a: runtime: O(1) [1], size: O(1) [0] activate: runtime: O(1) [9], size: O(n1) [1 + z]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, 1)