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:

a__f(X, X) → a__f(a, b)
a__b → a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__b → b

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:

a__f(X, X) → a__f(a, b) [1]
a__b → a [1]
mark(f(X1, X2)) → a__f(mark(X1), X2) [1]
mark(b) → a__b [1]
mark(a) → a [1]
a__f(X1, X2) → f(X1, X2) [1]
a__b → b [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:

a__f(X, X) → a__f(a, b) [1]
a__b → a [1]
mark(f(X1, X2)) → a__f(mark(X1), X2) [1]
mark(b) → a__b [1]
mark(a) → a [1]
a__f(X1, X2) → f(X1, X2) [1]
a__b → b [1]

The TRS has the following type information:

a__f :: a:b:f → a:b:f → a:b:f a :: a:b:f b :: a:b:f a__b :: a:b:f mark :: a:b:f → a:b:f f :: a:b:f → a:b:f → a:b:f

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:
none

(c) The following functions are completely defined:

mark
a__b
a__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: a__f(X, X) → a__f(a, b) [1] a__b → a [1] mark(f(X1, X2)) → a__f(mark(X1), X2) [1] mark(b) → a__b [1] mark(a) → a [1] a__f(X1, X2) → f(X1, X2) [1] a__b → b [1] The TRS has the following type information: a__f :: a:b:f → a:b:f → a:b:f a :: a:b:f b :: a:b:f a__b :: a:b:f mark :: a:b:f → a:b:f f :: a:b:f → a:b:f → a:b:f 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: a__f(X, X) → a__f(a, b) [1] a__b → a [1] mark(f(f(X1', X2'), X2)) → a__f(a__f(mark(X1'), X2'), X2) [2] mark(f(b, X2)) → a__f(a__b, X2) [2] mark(f(a, X2)) → a__f(a, X2) [2] mark(b) → a__b [1] mark(a) → a [1] a__f(X1, X2) → f(X1, X2) [1] a__b → b [1] The TRS has the following type information: a__f :: a:b:f → a:b:f → a:b:f a :: a:b:f b :: a:b:f a__b :: a:b:f mark :: a:b:f → a:b:f f :: a:b:f → a:b:f → a:b:f 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:

a => 0
b => 1

(10) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' = X, X >= 0, z = X
a__f(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 2 }→ a__f(a__b, X2) :|: z = 1 + 1 + X2, X2 >= 0
mark(z) -{ 2 }→ a__f(0, X2) :|: z = 1 + 0 + X2, X2 >= 0
mark(z) -{ 1 }→ a__b :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0

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

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

a__b -{ 1 }→ 0 :|:
a__b -{ 1 }→ 1 :|:

(12) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' = X, X >= 0, z = X
a__f(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, X2) :|: z = 1 + 1 + X2, X2 >= 0
mark(z) -{ 2 }→ a__f(0, X2) :|: z = 1 + 0 + X2, X2 >= 0
mark(z) -{ 3 }→ a__f(0, X2) :|: z = 1 + 1 + X2, X2 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

(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__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, z - 2) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__f(0, z - 2) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0, z - 1) :|: z - 1 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

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

Found the following analysis order by SCC decomposition:

{ a__b } { a__f } { mark }

(16) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, z - 2) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__f(0, z - 2) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0, z - 1) :|: z - 1 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

Function symbols to be analyzed: {a__b}, {a__f}, {mark}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, z - 2) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__f(0, z - 2) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0, z - 1) :|: z - 1 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

Function symbols to be analyzed: {a__b}, {a__f}, {mark}
Previous analysis results are:

a__b: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using CoFloCo for: a__b
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__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, z - 2) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__f(0, z - 2) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0, z - 1) :|: z - 1 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:

a__b: runtime: O(1) [1], size: O(1) [1]

(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__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, z - 2) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__f(0, z - 2) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0, z - 1) :|: z - 1 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:

a__b: runtime: O(1) [1], size: O(1) [1]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, z - 2) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__f(0, z - 2) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0, z - 1) :|: z - 1 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:

a__b: runtime: O(1) [1], size: O(1) [1] a__f: runtime: ?, size: O(n1) [2 + z + z']

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

Computed RUNTIME bound using CoFloCo for: a__f
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:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 1 }→ a__f(0, 1) :|: z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 3 }→ a__f(1, z - 2) :|: z - 2 >= 0
mark(z) -{ 3 }→ a__f(0, z - 2) :|: z - 2 >= 0
mark(z) -{ 2 }→ a__f(0, z - 1) :|: z - 1 >= 0
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

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

a__b: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [2], size: O(n1) [2 + 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:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * 1 + 2, z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * (z - 1) + 2, z - 1 >= 0
mark(z) -{ 5 }→ s'' :|: s'' >= 0, s'' <= 1 * 0 + 1 * (z - 2) + 2, z - 2 >= 0
mark(z) -{ 5 }→ s1 :|: s1 >= 0, s1 <= 1 * 1 + 1 * (z - 2) + 2, z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

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

a__b: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [2], size: O(n1) [2 + z + z']

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * 1 + 2, z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * (z - 1) + 2, z - 1 >= 0
mark(z) -{ 5 }→ s'' :|: s'' >= 0, s'' <= 1 * 0 + 1 * (z - 2) + 2, z - 2 >= 0
mark(z) -{ 5 }→ s1 :|: s1 >= 0, s1 <= 1 * 1 + 1 * (z - 2) + 2, z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

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

a__b: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [2], size: O(n1) [2 + z + z'] mark: runtime: ?, size: O(n1) [2·z]

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

Computed RUNTIME bound using KoAT for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 17 + 6·z

(32) Obligation:

Complexity RNTS consisting of the following rules:

a__b -{ 1 }→ 1 :|:
a__b -{ 1 }→ 0 :|:
a__f(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * 1 + 2, z' >= 0, z = z'
a__f(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
mark(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * (z - 1) + 2, z - 1 >= 0
mark(z) -{ 5 }→ s'' :|: s'' >= 0, s'' <= 1 * 0 + 1 * (z - 2) + 2, z - 2 >= 0
mark(z) -{ 5 }→ s1 :|: s1 >= 0, s1 <= 1 * 1 + 1 * (z - 2) + 2, z - 2 >= 0
mark(z) -{ 2 }→ a__f(a__f(mark(X1'), X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2
mark(z) -{ 2 }→ 1 :|: z = 1
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 2 }→ 0 :|: z = 1

Function symbols to be analyzed:
Previous analysis results are:

a__b: runtime: O(1) [1], size: O(1) [1] a__f: runtime: O(1) [2], size: O(n1) [2 + z + z'] mark: runtime: O(n1) [17 + 6·z], size: O(n1) [2·z]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^1)