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:

f(0) → 1
f(s(x)) → g(f(x))
g(x) → +(x, s(x))
f(s(x)) → +(f(x), s(f(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:

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

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

The TRS has the following type information:

f :: 0:1:s:+ → 0:1:s:+ 0 :: 0:1:s:+ 1 :: 0:1:s:+ s :: 0:1:s:+ → 0:1:s:+ g :: 0:1:s:+ → 0:1:s:+ + :: 0:1:s:+ → 0:1:s:+ → 0:1: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:
none

(c) The following functions are completely defined:

f
g

Due to the following rules being added:

f(v0) → null_f [0]

And the following fresh constants:

null_f

(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(0) → 1 [1] f(s(x)) → g(f(x)) [1] g(x) → +(x, s(x)) [1] f(s(x)) → +(f(x), s(f(x))) [1] f(v0) → null_f [0] The TRS has the following type information: f :: 0:1:s:+:null_f → 0:1:s:+:null_f 0 :: 0:1:s:+:null_f 1 :: 0:1:s:+:null_f s :: 0:1:s:+:null_f → 0:1:s:+:null_f g :: 0:1:s:+:null_f → 0:1:s:+:null_f + :: 0:1:s:+:null_f → 0:1:s:+:null_f → 0:1:s:+:null_f null_f :: 0:1:s:+:null_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: f(0) → 1 [1] f(s(0)) → g(1) [2] f(s(s(x'))) → g(g(f(x'))) [2] f(s(s(x''))) → g(+(f(x''), s(f(x'')))) [2] f(s(x)) → g(null_f) [1] g(x) → +(x, s(x)) [1] f(s(x)) → +(f(x), s(f(x))) [1] f(v0) → null_f [0] The TRS has the following type information: f :: 0:1:s:+:null_f → 0:1:s:+:null_f 0 :: 0:1:s:+:null_f 1 :: 0:1:s:+:null_f s :: 0:1:s:+:null_f → 0:1:s:+:null_f g :: 0:1:s:+:null_f → 0:1:s:+:null_f + :: 0:1:s:+:null_f → 0:1:s:+:null_f → 0:1:s:+:null_f null_f :: 0:1:s:+:null_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:

0 => 0
1 => 1
null_f => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(g(f(x'))) :|: x' >= 0, z = 1 + (1 + x')
f(z) -{ 2 }→ g(1) :|: z = 1 + 0
f(z) -{ 1 }→ g(0) :|: x >= 0, z = 1 + x
f(z) -{ 2 }→ g(1 + f(x'') + (1 + f(x''))) :|: x'' >= 0, z = 1 + (1 + x'')
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
f(z) -{ 1 }→ 1 + f(x) + (1 + f(x)) :|: x >= 0, z = 1 + x
g(z) -{ 1 }→ 1 + x + (1 + x) :|: x >= 0, z = x

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

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(g(f(x'))) :|: x' >= 0, z = 1 + (1 + x')
f(z) -{ 2 }→ g(1 + f(x'') + (1 + f(x''))) :|: x'' >= 0, z = 1 + (1 + x'')
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: x >= 0, z = 1 + x, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(x) + (1 + f(x)) :|: x >= 0, z = 1 + x
g(z) -{ 1 }→ 1 + x + (1 + x) :|: 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) -{ 2 }→ g(g(f(z - 2))) :|: z - 2 >= 0
f(z) -{ 2 }→ g(1 + f(z - 2) + (1 + f(z - 2))) :|: z - 2 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: z - 1 >= 0, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(z - 1) + (1 + f(z - 1)) :|: z - 1 >= 0
g(z) -{ 1 }→ 1 + z + (1 + z) :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ g } { f }

(16) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(g(f(z - 2))) :|: z - 2 >= 0
f(z) -{ 2 }→ g(1 + f(z - 2) + (1 + f(z - 2))) :|: z - 2 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: z - 1 >= 0, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(z - 1) + (1 + f(z - 1)) :|: z - 1 >= 0
g(z) -{ 1 }→ 1 + z + (1 + z) :|: z >= 0

Function symbols to be analyzed: {g}, {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: 2 + 2·z

(18) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(g(f(z - 2))) :|: z - 2 >= 0
f(z) -{ 2 }→ g(1 + f(z - 2) + (1 + f(z - 2))) :|: z - 2 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: z - 1 >= 0, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(z - 1) + (1 + f(z - 1)) :|: z - 1 >= 0
g(z) -{ 1 }→ 1 + z + (1 + z) :|: z >= 0

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

g: runtime: ?, size: O(n1) [2 + 2·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) -{ 2 }→ g(g(f(z - 2))) :|: z - 2 >= 0
f(z) -{ 2 }→ g(1 + f(z - 2) + (1 + f(z - 2))) :|: z - 2 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: z - 1 >= 0, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(z - 1) + (1 + f(z - 1)) :|: z - 1 >= 0
g(z) -{ 1 }→ 1 + z + (1 + z) :|: z >= 0

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

g: runtime: O(1) [1], size: O(n1) [2 + 2·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) -{ 2 }→ g(g(f(z - 2))) :|: z - 2 >= 0
f(z) -{ 2 }→ g(1 + f(z - 2) + (1 + f(z - 2))) :|: z - 2 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: z - 1 >= 0, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(z - 1) + (1 + f(z - 1)) :|: z - 1 >= 0
g(z) -{ 1 }→ 1 + z + (1 + z) :|: z >= 0

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

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(g(f(z - 2))) :|: z - 2 >= 0
f(z) -{ 2 }→ g(1 + f(z - 2) + (1 + f(z - 2))) :|: z - 2 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: z - 1 >= 0, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(z - 1) + (1 + f(z - 1)) :|: z - 1 >= 0
g(z) -{ 1 }→ 1 + z + (1 + z) :|: z >= 0

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

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

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(g(f(z - 2))) :|: z - 2 >= 0
f(z) -{ 2 }→ g(1 + f(z - 2) + (1 + f(z - 2))) :|: z - 2 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 1 + x + (1 + x) :|: z = 1 + 0, x >= 0, 1 = x
f(z) -{ 2 }→ 1 + x' + (1 + x') :|: z - 1 >= 0, x' >= 0, 0 = x'
f(z) -{ 1 }→ 1 + f(z - 1) + (1 + f(z - 1)) :|: z - 1 >= 0
g(z) -{ 1 }→ 1 + z + (1 + z) :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:

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

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, EXP)