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))
double(X) → +(X, X)
f(0, s(0), X) → f(X, double(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]
double(X) → +(X, X) [1]
f(0, s(0), X) → f(X, double(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]
double(X) → plus(X, X) [1]
f(0, s(0), X) → f(X, double(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]
double(X) → plus(X, X) [1]
f(0, s(0), X) → f(X, double(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 double :: 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:

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

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

(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:

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

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

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

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

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

double(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z, z >= 0
f(z, z', z'') -{ 3 + z'' }→ f(z'', s'', z'') :|: s'' >= 0, s'' <= 1 * z'' + 1 * z'', z'' >= 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}, {double}
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:

double(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z, z >= 0
f(z, z', z'') -{ 3 + z'' }→ f(z'', s'', z'') :|: s'' >= 0, s'' <= 1 * z'' + 1 * z'', z'' >= 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}, {double}
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: 3 + z''

(32) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z, z >= 0
f(z, z', z'') -{ 3 + z'' }→ f(z'', s'', z'') :|: s'' >= 0, s'' <= 1 * z'' + 1 * z'', z'' >= 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: {double}
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) [3 + z''], 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:

double(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z, z >= 0
f(z, z', z'') -{ 6 + 2·z'' }→ s1 :|: s1 >= 0, s1 <= 0, s'' >= 0, s'' <= 1 * z'' + 1 * z'', z'' >= 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: {double}
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) [3 + z''], size: O(1) [0]

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

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z, z >= 0
f(z, z', z'') -{ 6 + 2·z'' }→ s1 :|: s1 >= 0, s1 <= 0, s'' >= 0, s'' <= 1 * z'' + 1 * z'', z'' >= 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: {double}
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) [3 + z''], size: O(1) [0] double: runtime: ?, size: O(n1) [2·z]

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

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z, z >= 0
f(z, z', z'') -{ 6 + 2·z'' }→ s1 :|: s1 >= 0, s1 <= 0, s'' >= 0, s'' <= 1 * z'' + 1 * z'', z'' >= 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) [3 + z''], size: O(1) [0] double: runtime: O(n1) [2 + z], size: O(n1) [2·z]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^1)