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:

+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
f(0, s(0), X) → f(X, +(X, X), X)
g(X, Y) → X
g(X, Y) → Y

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:

+(X, 0) → X [1]
+(X, s(Y)) → s(+(X, Y)) [1]
f(0, s(0), X) → f(X, +(X, X), X) [1]
g(X, Y) → X [1]
g(X, Y) → Y [1]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

(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:

plus(X, 0) → X [1]
plus(X, s(Y)) → s(plus(X, Y)) [1]
f(0, s(0), X) → f(X, plus(X, X), X) [1]
g(X, Y) → X [1]
g(X, Y) → Y [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:

plus(X, 0) → X [1]
plus(X, s(Y)) → s(plus(X, Y)) [1]
f(0, s(0), X) → f(X, plus(X, X), X) [1]
g(X, Y) → X [1]
g(X, Y) → Y [1]

The TRS has the following type information:

plus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s f :: 0:s → 0:s → 0:s → f g :: g → g → 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:

f
g

(c) The following functions are completely defined:

plus

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: plus(X, 0) → X [1] plus(X, s(Y)) → s(plus(X, Y)) [1] f(0, s(0), X) → f(X, plus(X, X), X) [1] g(X, Y) → X [1] g(X, Y) → Y [1] The TRS has the following type information: plus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s f :: 0:s → 0:s → 0:s → f g :: g → g → g const :: f const1 :: g 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: plus(X, 0) → X [1] plus(X, s(Y)) → s(plus(X, Y)) [1] f(0, s(0), 0) → f(0, 0, 0) [2] f(0, s(0), s(Y')) → f(s(Y'), s(plus(s(Y'), Y')), s(Y')) [2] g(X, Y) → X [1] g(X, Y) → Y [1] The TRS has the following type information: plus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s f :: 0:s → 0:s → 0:s → f g :: g → g → g const :: f const1 :: g 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:

0 => 0
const => 0
const1 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

Found the following analysis order by SCC decomposition:

{ g } { plus } { f }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

(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: z + z'

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

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

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

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

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

(23) 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'

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(25) 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'

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

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

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

Computed SIZE bound using CoFloCo for: f
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:

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

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

g: runtime: O(1) [1], size: O(n1) [z + z'] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] f: runtime: ?, size: O(1) [0]

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

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

g: runtime: O(1) [1], size: O(n1) [z + z'] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] f: runtime: O(n1) [2 + z''], size: O(1) [0]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^1)