The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

and(tt, X) → activate(X)
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))
x(N, 0) → 0
x(N, s(M)) → plus(x(N, M), N)
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^2).


The TRS R consists of the following rules:

and(tt, X) → activate(X) [1]
plus(N, 0) → N [1]
plus(N, s(M)) → s(plus(N, M)) [1]
x(N, 0) → 0 [1]
x(N, s(M)) → plus(x(N, M), N) [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:

and(tt, X) → activate(X) [1]
plus(N, 0) → N [1]
plus(N, s(M)) → s(plus(N, M)) [1]
x(N, 0) → 0 [1]
x(N, s(M)) → plus(x(N, M), N) [1]
activate(X) → X [1]

The TRS has the following type information:

and :: tt → and:activate → and:activate tt :: tt activate :: and:activate → and:activate plus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s x :: 0:s → 0:s → 0:s

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:

and
activate

(c) The following functions are completely defined:

x
plus

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: and(tt, X) → activate(X) [1] plus(N, 0) → N [1] plus(N, s(M)) → s(plus(N, M)) [1] x(N, 0) → 0 [1] x(N, s(M)) → plus(x(N, M), N) [1] activate(X) → X [1] The TRS has the following type information: and :: tt → and:activate → and:activate tt :: tt activate :: and:activate → and:activate plus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s x :: 0:s → 0:s → 0:s const :: and:activate 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: and(tt, X) → activate(X) [1] plus(N, 0) → N [1] plus(N, s(M)) → s(plus(N, M)) [1] x(N, 0) → 0 [1] x(N, s(0)) → plus(0, N) [2] x(N, s(s(M'))) → plus(plus(x(N, M'), N), N) [2] activate(X) → X [1] The TRS has the following type information: and :: tt → and:activate → and:activate tt :: tt activate :: and:activate → and:activate plus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s x :: 0:s → 0:s → 0:s const :: and:activate 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:

tt => 0
0 => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
and(z, z') -{ 1 }→ activate(X) :|: z' = X, X >= 0, z = 0
plus(z, z') -{ 1 }→ N :|: z = N, z' = 0, N >= 0
plus(z, z') -{ 1 }→ 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0
x(z, z') -{ 2 }→ plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0
x(z, z') -{ 2 }→ plus(0, N) :|: z = N, z' = 1 + 0, N >= 0
x(z, z') -{ 1 }→ 0 :|: z = N, z' = 0, N >= 0

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

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

activate(z) -{ 1 }→ X :|: X >= 0, z = X

(12) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
and(z, z') -{ 2 }→ X' :|: z' = X, X >= 0, z = 0, X' >= 0, X = X'
plus(z, z') -{ 1 }→ N :|: z = N, z' = 0, N >= 0
plus(z, z') -{ 1 }→ 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0
x(z, z') -{ 2 }→ plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0
x(z, z') -{ 2 }→ plus(0, N) :|: z = N, z' = 1 + 0, N >= 0
x(z, z') -{ 1 }→ 0 :|: z = N, z' = 0, N >= 0

(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
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

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

Found the following analysis order by SCC decomposition:

{ activate } { and } { plus } { x }

(16) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {and}, {plus}, {x}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {and}, {plus}, {x}
Previous analysis results are:

activate: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using CoFloCo for: activate
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
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {and}, {plus}, {x}
Previous analysis results are:

activate: runtime: O(1) [1], size: O(n1) [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
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {and}, {plus}, {x}
Previous analysis results are:

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {and}, {plus}, {x}
Previous analysis results are:

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: ?, size: O(n1) [z']

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

Computed RUNTIME bound using CoFloCo for: and
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(26) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: O(1) [2], size: O(n1) [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
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: O(1) [2], size: O(n1) [z']

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: O(1) [2], size: O(n1) [z'] plus: runtime: ?, size: O(n1) [z + z']

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

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 2 }→ plus(0, z) :|: z' = 1 + 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

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

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: O(1) [2], size: O(n1) [z'] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z']

(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:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 + z' }→ 1 + s :|: s >= 0, s <= 1 * z + 1 * (z' - 1), z' - 1 >= 0, z >= 0
x(z, z') -{ 3 + z }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * z, z' = 1 + 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

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

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: O(1) [2], size: O(n1) [z'] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z']

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

Computed SIZE bound using KoAT for: x
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z + 2·z·z'

(36) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 + z' }→ 1 + s :|: s >= 0, s <= 1 * z + 1 * (z' - 1), z' - 1 >= 0, z >= 0
x(z, z') -{ 3 + z }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * z, z' = 1 + 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

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

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: O(1) [2], size: O(n1) [z'] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] x: runtime: ?, size: O(n2) [z + 2·z·z']

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

Computed RUNTIME bound using PUBS for: x
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 3 + z + 2·z·z' + 4·z'

(38) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
and(z, z') -{ 2 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z' = 0, z >= 0
plus(z, z') -{ 1 + z' }→ 1 + s :|: s >= 0, s <= 1 * z + 1 * (z' - 1), z' - 1 >= 0, z >= 0
x(z, z') -{ 3 + z }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * z, z' = 1 + 0, z >= 0
x(z, z') -{ 2 }→ plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
x(z, z') -{ 1 }→ 0 :|: z' = 0, z >= 0

Function symbols to be analyzed:
Previous analysis results are:

activate: runtime: O(1) [1], size: O(n1) [z] and: runtime: O(1) [2], size: O(n1) [z'] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] x: runtime: O(n2) [3 + z + 2·z·z' + 4·z'], size: O(n2) [z + 2·z·z']

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^2)