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:

f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

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:

f(a, empty) → g(a, empty) [1]
f(a, cons(x, k)) → f(cons(x, a), k) [1]
g(empty, d) → d [1]
g(cons(x, k), d) → g(k, cons(x, d)) [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(a, empty) → g(a, empty) [1]
f(a, cons(x, k)) → f(cons(x, a), k) [1]
g(empty, d) → d [1]
g(cons(x, k), d) → g(k, cons(x, d)) [1]

The TRS has the following type information:

f :: empty:cons → empty:cons → empty:cons empty :: empty:cons g :: empty:cons → empty:cons → empty:cons cons :: a → empty:cons → empty:cons

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:

f
g

(c) The following functions are completely defined:
none

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: f(a, empty) → g(a, empty) [1] f(a, cons(x, k)) → f(cons(x, a), k) [1] g(empty, d) → d [1] g(cons(x, k), d) → g(k, cons(x, d)) [1] The TRS has the following type information: f :: empty:cons → empty:cons → empty:cons empty :: empty:cons g :: empty:cons → empty:cons → empty:cons cons :: a → empty:cons → empty:cons const :: a 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(a, empty) → g(a, empty) [1] f(a, cons(x, k)) → f(cons(x, a), k) [1] g(empty, d) → d [1] g(cons(x, k), d) → g(k, cons(x, d)) [1] The TRS has the following type information: f :: empty:cons → empty:cons → empty:cons empty :: empty:cons g :: empty:cons → empty:cons → empty:cons cons :: a → empty:cons → empty:cons const :: a 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:

empty => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 1 }→ g(a, 0) :|: z = a, a >= 0, z' = 0
f(z, z') -{ 1 }→ f(1 + x + a, k) :|: z = a, a >= 0, x >= 0, z' = 1 + x + k, k >= 0
g(z, z') -{ 1 }→ d :|: z' = d, d >= 0, z = 0
g(z, z') -{ 1 }→ g(k, 1 + x + d) :|: z' = d, x >= 0, d >= 0, k >= 0, z = 1 + x + k

(11) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(12) Obligation:

Complexity RNTS consisting of the following rules:

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

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

Found the following analysis order by SCC decomposition:

{ g } { f }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

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

(15) 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'

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

Computed RUNTIME 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:

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

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

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

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(23) 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 + 2·z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

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

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)