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

(0) Obligation:

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


The TRS R consists of the following rules:

fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(x))) → +(fib(s(x)), fib(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, EXP).


The TRS R consists of the following rules:

fib(0) → 0 [1]
fib(s(0)) → s(0) [1]
fib(s(s(x))) → +(fib(s(x)), fib(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:

fib(0) → 0 [1]
fib(s(0)) → s(0) [1]
fib(s(s(x))) → +(fib(s(x)), fib(x)) [1]

The TRS has the following type information:

fib :: 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ + :: 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:

fib

(c) The following functions are completely defined:
none

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: fib(0) → 0 [1] fib(s(0)) → s(0) [1] fib(s(s(x))) → +(fib(s(x)), fib(x)) [1] The TRS has the following type information: fib :: 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ 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: fib(0) → 0 [1] fib(s(0)) → s(0) [1] fib(s(s(x))) → +(fib(s(x)), fib(x)) [1] The TRS has the following type information: fib :: 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ 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:

0 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

fib(z) -{ 1 }→ 0 :|: z = 0
fib(z) -{ 1 }→ 1 + 0 :|: z = 1 + 0
fib(z) -{ 1 }→ 1 + fib(1 + x) + fib(x) :|: x >= 0, z = 1 + (1 + x)

(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:

fib(z) -{ 1 }→ 0 :|: z = 0
fib(z) -{ 1 }→ 1 + 0 :|: z = 1 + 0
fib(z) -{ 1 }→ 1 + fib(1 + (z - 2)) + fib(z - 2) :|: z - 2 >= 0

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

Found the following analysis order by SCC decomposition:

{ fib }

(14) Obligation:

Complexity RNTS consisting of the following rules:

fib(z) -{ 1 }→ 0 :|: z = 0
fib(z) -{ 1 }→ 1 + 0 :|: z = 1 + 0
fib(z) -{ 1 }→ 1 + fib(1 + (z - 2)) + fib(z - 2) :|: z - 2 >= 0

Function symbols to be analyzed: {fib}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: fib
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(16) Obligation:

Complexity RNTS consisting of the following rules:

fib(z) -{ 1 }→ 0 :|: z = 0
fib(z) -{ 1 }→ 1 + 0 :|: z = 1 + 0
fib(z) -{ 1 }→ 1 + fib(1 + (z - 2)) + fib(z - 2) :|: z - 2 >= 0

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

fib: runtime: ?, size: EXP

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

Computed RUNTIME bound using KoAT for: fib
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(18) Obligation:

Complexity RNTS consisting of the following rules:

fib(z) -{ 1 }→ 0 :|: z = 0
fib(z) -{ 1 }→ 1 + 0 :|: z = 1 + 0
fib(z) -{ 1 }→ 1 + fib(1 + (z - 2)) + fib(z - 2) :|: z - 2 >= 0

Function symbols to be analyzed:
Previous analysis results are:

fib: runtime: EXP, size: EXP

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, EXP)