The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, 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, n^2).


The TRS R consists of the following rules:

f(0) → 1 [1]
f(s(x)) → g(x, s(x)) [1]
g(0, y) → y [1]
g(s(x), y) → g(x, +(y, s(x))) [1]
+(x, 0) → x [1]
+(x, s(y)) → s(+(x, y)) [1]
g(s(x), y) → g(x, s(+(y, x))) [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^2).


The TRS R consists of the following rules:

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

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

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:

plus

Due to the following rules being added:

plus(v0, v1) → null_plus [0]

And the following fresh constants:

null_plus

(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(x)) → g(x, s(x)) [1] g(0, y) → y [1] g(s(x), y) → g(x, plus(y, s(x))) [1] plus(x, 0) → x [1] plus(x, s(y)) → s(plus(x, y)) [1] g(s(x), y) → g(x, s(plus(y, x))) [1] plus(v0, v1) → null_plus [0] The TRS has the following type information: f :: 0:1:s:null_plus → 0:1:s:null_plus 0 :: 0:1:s:null_plus 1 :: 0:1:s:null_plus s :: 0:1:s:null_plus → 0:1:s:null_plus g :: 0:1:s:null_plus → 0:1:s:null_plus → 0:1:s:null_plus plus :: 0:1:s:null_plus → 0:1:s:null_plus → 0:1:s:null_plus null_plus :: 0:1:s:null_plus 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: f(0) → 1 [1] f(s(x)) → g(x, s(x)) [1] g(0, y) → y [1] g(s(x), y) → g(x, s(plus(y, x))) [2] g(s(x), y) → g(x, null_plus) [1] plus(x, 0) → x [1] plus(x, s(y)) → s(plus(x, y)) [1] g(s(0), y) → g(0, s(y)) [2] g(s(s(y')), y) → g(s(y'), s(s(plus(y, y')))) [2] g(s(x), y) → g(x, s(null_plus)) [1] plus(v0, v1) → null_plus [0] The TRS has the following type information: f :: 0:1:s:null_plus → 0:1:s:null_plus 0 :: 0:1:s:null_plus 1 :: 0:1:s:null_plus s :: 0:1:s:null_plus → 0:1:s:null_plus g :: 0:1:s:null_plus → 0:1:s:null_plus → 0:1:s:null_plus plus :: 0:1:s:null_plus → 0:1:s:null_plus → 0:1:s:null_plus null_plus :: 0:1:s:null_plus 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
1 => 1
null_plus => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 1 }→ g(x, 1 + x) :|: x >= 0, z = 1 + x
f(z) -{ 1 }→ 1 :|: z = 0
g(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
g(z, z') -{ 1 }→ g(x, 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y
g(z, z') -{ 2 }→ g(x, 1 + plus(y, x)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
g(z, z') -{ 1 }→ g(x, 1 + 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y
g(z, z') -{ 2 }→ g(0, 1 + y) :|: z = 1 + 0, y >= 0, z' = y
g(z, z') -{ 2 }→ g(1 + y', 1 + (1 + plus(y, y'))) :|: y >= 0, z = 1 + (1 + y'), y' >= 0, z' = y
plus(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
plus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 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) -{ 1 }→ g(z - 1, 1 + (z - 1)) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
g(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
g(z, z') -{ 2 }→ g(0, 1 + z') :|: z = 1 + 0, z' >= 0
g(z, z') -{ 1 }→ g(z - 1, 0) :|: z - 1 >= 0, z' >= 0
g(z, z') -{ 2 }→ g(z - 1, 1 + plus(z', z - 1)) :|: z - 1 >= 0, z' >= 0
g(z, z') -{ 1 }→ g(z - 1, 1 + 0) :|: z - 1 >= 0, z' >= 0
g(z, z') -{ 2 }→ g(1 + (z - 2), 1 + (1 + plus(z', z - 2))) :|: z' >= 0, z - 2 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ plus } { g } { f }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

(17) 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'

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

(19) 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'

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

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

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

Computed SIZE bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z² + z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

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

plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] g: runtime: ?, size: O(n2) [z2 + z']

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

Computed RUNTIME bound using PUBS for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + 2·z + z²

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

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

plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] g: runtime: O(n2) [1 + 2·z + z2], size: O(n2) [z2 + 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:

f(z) -{ 1 + z² }→ s1 :|: s1 >= 0, s1 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + (z - 1)), z - 1 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
g(z, z') -{ 2 + z + z² }→ s2 :|: s2 >= 0, s2 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + s), s >= 0, s <= 1 * z' + 1 * (z - 1), z - 1 >= 0, z' >= 0
g(z, z') -{ 1 + z² }→ s3 :|: s3 >= 0, s3 <= 1 * ((z - 1) * (z - 1)) + 1 * 0, z - 1 >= 0, z' >= 0
g(z, z') -{ 3 }→ s4 :|: s4 >= 0, s4 <= 1 * (0 * 0) + 1 * (1 + z'), z = 1 + 0, z' >= 0
g(z, z') -{ 1 + z + z² }→ s5 :|: s5 >= 0, s5 <= 1 * ((1 + (z - 2)) * (1 + (z - 2))) + 1 * (1 + (1 + s'')), s'' >= 0, s'' <= 1 * z' + 1 * (z - 2), z' >= 0, z - 2 >= 0
g(z, z') -{ 1 + z² }→ s6 :|: s6 >= 0, s6 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + 0), z - 1 >= 0, z' >= 0
g(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 1 + z' }→ 1 + s' :|: s' >= 0, s' <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0

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

plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] g: runtime: O(n2) [1 + 2·z + z2], size: O(n2) [z2 + z']

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

Computed SIZE bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2 + z + z²

(30) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 1 + z² }→ s1 :|: s1 >= 0, s1 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + (z - 1)), z - 1 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
g(z, z') -{ 2 + z + z² }→ s2 :|: s2 >= 0, s2 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + s), s >= 0, s <= 1 * z' + 1 * (z - 1), z - 1 >= 0, z' >= 0
g(z, z') -{ 1 + z² }→ s3 :|: s3 >= 0, s3 <= 1 * ((z - 1) * (z - 1)) + 1 * 0, z - 1 >= 0, z' >= 0
g(z, z') -{ 3 }→ s4 :|: s4 >= 0, s4 <= 1 * (0 * 0) + 1 * (1 + z'), z = 1 + 0, z' >= 0
g(z, z') -{ 1 + z + z² }→ s5 :|: s5 >= 0, s5 <= 1 * ((1 + (z - 2)) * (1 + (z - 2))) + 1 * (1 + (1 + s'')), s'' >= 0, s'' <= 1 * z' + 1 * (z - 2), z' >= 0, z - 2 >= 0
g(z, z') -{ 1 + z² }→ s6 :|: s6 >= 0, s6 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + 0), z - 1 >= 0, z' >= 0
g(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 1 + z' }→ 1 + s' :|: s' >= 0, s' <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0

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

plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] g: runtime: O(n2) [1 + 2·z + z2], size: O(n2) [z2 + z'] f: runtime: ?, size: O(n2) [2 + z + z2]

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

Computed RUNTIME bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2 + z²

(32) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 1 + z² }→ s1 :|: s1 >= 0, s1 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + (z - 1)), z - 1 >= 0
f(z) -{ 1 }→ 1 :|: z = 0
g(z, z') -{ 2 + z + z² }→ s2 :|: s2 >= 0, s2 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + s), s >= 0, s <= 1 * z' + 1 * (z - 1), z - 1 >= 0, z' >= 0
g(z, z') -{ 1 + z² }→ s3 :|: s3 >= 0, s3 <= 1 * ((z - 1) * (z - 1)) + 1 * 0, z - 1 >= 0, z' >= 0
g(z, z') -{ 3 }→ s4 :|: s4 >= 0, s4 <= 1 * (0 * 0) + 1 * (1 + z'), z = 1 + 0, z' >= 0
g(z, z') -{ 1 + z + z² }→ s5 :|: s5 >= 0, s5 <= 1 * ((1 + (z - 2)) * (1 + (z - 2))) + 1 * (1 + (1 + s'')), s'' >= 0, s'' <= 1 * z' + 1 * (z - 2), z' >= 0, z - 2 >= 0
g(z, z') -{ 1 + z² }→ s6 :|: s6 >= 0, s6 <= 1 * ((z - 1) * (z - 1)) + 1 * (1 + 0), z - 1 >= 0, z' >= 0
g(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 1 + z' }→ 1 + s' :|: s' >= 0, s' <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] g: runtime: O(n2) [1 + 2·z + z2], size: O(n2) [z2 + z'] f: runtime: O(n2) [2 + z2], size: O(n2) [2 + z + z2]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^2)