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) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Rewrite Strategy: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of g: f, g
The following defined symbols can occur below the 1th argument of g: f, g

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(g(x, y)) → g(f(x), f(y))

(2) 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) → b
f(c) → d
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(4) 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) → b [1]
f(c) → d [1]
f(h(x, y)) → g(h(y, f(x)), h(x, f(y))) [1]
g(x, x) → h(e, 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(a) → b [1]
f(c) → d [1]
f(h(x, y)) → g(h(y, f(x)), h(x, f(y))) [1]
g(x, x) → h(e, x) [1]

The TRS has the following type information:

f :: a:b:c:d:h:e → a:b:c:d:h:e a :: a:b:c:d:h:e b :: a:b:c:d:h:e c :: a:b:c:d:h:e d :: a:b:c:d:h:e h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e e :: a:b:c:d:h:e

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:
none

(c) The following functions are completely defined:

f
g

Due to the following rules being added:

f(v0) → null_f [0]
g(v0, v1) → null_g [0]

And the following fresh constants:

null_f, null_g

(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) → b [1] f(c) → d [1] f(h(x, y)) → g(h(y, f(x)), h(x, f(y))) [1] g(x, x) → h(e, x) [1] f(v0) → null_f [0] g(v0, v1) → null_g [0] The TRS has the following type information: f :: a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g a :: a:b:c:d:h:e:null_f:null_g b :: a:b:c:d:h:e:null_f:null_g c :: a:b:c:d:h:e:null_f:null_g d :: a:b:c:d:h:e:null_f:null_g h :: a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g g :: a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g e :: a:b:c:d:h:e:null_f:null_g null_f :: a:b:c:d:h:e:null_f:null_g null_g :: a:b:c:d:h:e:null_f:null_g 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(a) → b [1] f(c) → d [1] f(h(a, a)) → g(h(a, b), h(a, b)) [3] f(h(a, c)) → g(h(c, b), h(a, d)) [3] f(h(a, h(x'', y''))) → g(h(h(x'', y''), b), h(a, g(h(y'', f(x'')), h(x'', f(y''))))) [3] f(h(a, y)) → g(h(y, b), h(a, null_f)) [2] f(h(c, a)) → g(h(a, d), h(c, b)) [3] f(h(c, c)) → g(h(c, d), h(c, d)) [3] f(h(c, h(x1, y1))) → g(h(h(x1, y1), d), h(c, g(h(y1, f(x1)), h(x1, f(y1))))) [3] f(h(c, y)) → g(h(y, d), h(c, null_f)) [2] f(h(h(x', y'), a)) → g(h(a, g(h(y', f(x')), h(x', f(y')))), h(h(x', y'), b)) [3] f(h(h(x', y'), c)) → g(h(c, g(h(y', f(x')), h(x', f(y')))), h(h(x', y'), d)) [3] f(h(h(x', y'), h(x2, y2))) → g(h(h(x2, y2), g(h(y', f(x')), h(x', f(y')))), h(h(x', y'), g(h(y2, f(x2)), h(x2, f(y2))))) [3] f(h(h(x', y'), y)) → g(h(y, g(h(y', f(x')), h(x', f(y')))), h(h(x', y'), null_f)) [2] f(h(x, a)) → g(h(a, null_f), h(x, b)) [2] f(h(x, c)) → g(h(c, null_f), h(x, d)) [2] f(h(x, h(x3, y3))) → g(h(h(x3, y3), null_f), h(x, g(h(y3, f(x3)), h(x3, f(y3))))) [2] f(h(x, y)) → g(h(y, null_f), h(x, null_f)) [1] g(x, x) → h(e, x) [1] f(v0) → null_f [0] g(v0, v1) → null_g [0] The TRS has the following type information: f :: a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g a :: a:b:c:d:h:e:null_f:null_g b :: a:b:c:d:h:e:null_f:null_g c :: a:b:c:d:h:e:null_f:null_g d :: a:b:c:d:h:e:null_f:null_g h :: a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g g :: a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g → a:b:c:d:h:e:null_f:null_g e :: a:b:c:d:h:e:null_f:null_g null_f :: a:b:c:d:h:e:null_f:null_g null_g :: a:b:c:d:h:e:null_f:null_g 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:

a => 0
b => 1
c => 2
d => 3
e => 4
null_f => 0
null_g => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 2 }→ g(1 + y + 3, 1 + 2 + 0) :|: y >= 0, z = 1 + 2 + y
f(z) -{ 2 }→ g(1 + y + 1, 1 + 0 + 0) :|: y >= 0, z = 1 + 0 + y
f(z) -{ 1 }→ g(1 + y + 0, 1 + x + 0) :|: z = 1 + x + y, x >= 0, y >= 0
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 2 + 3, 1 + 2 + 3) :|: z = 1 + 2 + 2
f(z) -{ 3 }→ g(1 + 2 + 1, 1 + 0 + 3) :|: z = 1 + 0 + 2
f(z) -{ 2 }→ g(1 + 2 + 0, 1 + x + 3) :|: x >= 0, z = 1 + x + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + 0 + 3, 1 + 2 + 1) :|: z = 1 + 2 + 0
f(z) -{ 3 }→ g(1 + 0 + 1, 1 + 0 + 1) :|: z = 1 + 0 + 0
f(z) -{ 2 }→ g(1 + 0 + 0, 1 + x + 1) :|: x >= 0, z = 1 + x + 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
g(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
g(z, z') -{ 1 }→ 1 + 4 + x :|: z' = x, x >= 0, z = x

(13) InliningProof (UPPER BOUND(ID) transformation)

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

g(z, z') -{ 1 }→ 1 + 4 + x :|: z' = x, x >= 0, z = x
g(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

(14) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: y >= 0, z = 1 + 0 + y, v0 >= 0, v1 >= 0, 1 + y + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: y >= 0, z = 1 + 2 + y, v0 >= 0, v1 >= 0, 1 + y + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: x >= 0, z = 1 + x + 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + x + 1 = v1
f(z) -{ 2 }→ 0 :|: x >= 0, z = 1 + x + 2, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + x + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
g(z, z') -{ 1 }→ 1 + 4 + x :|: z' = x, x >= 0, z = x

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 1) + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 3) + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + (z - 1) + 1 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + (z - 3) + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
g(z, z') -{ 1 }→ 1 + 4 + z' :|: z' >= 0, z = z'

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

Found the following analysis order by SCC decomposition:

{ g } { f }

(18) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 1) + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 3) + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + (z - 1) + 1 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + (z - 3) + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
g(z, z') -{ 1 }→ 1 + 4 + z' :|: z' >= 0, z = z'

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

(19) 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: 5 + z'

(20) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 1) + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 3) + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + (z - 1) + 1 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + (z - 3) + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
g(z, z') -{ 1 }→ 1 + 4 + z' :|: z' >= 0, z = z'

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

g: runtime: ?, size: O(n1) [5 + z']

(21) 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

(22) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 1) + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 3) + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + (z - 1) + 1 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + (z - 3) + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
g(z, z') -{ 1 }→ 1 + 4 + z' :|: z' >= 0, z = z'

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

g: runtime: O(1) [1], size: O(n1) [5 + z']

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 1) + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 3) + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + (z - 1) + 1 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + (z - 3) + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
g(z, z') -{ 1 }→ 1 + 4 + z' :|: z' >= 0, z = z'

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

g: runtime: O(1) [1], size: O(n1) [5 + z']

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

Computed SIZE bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 6·z

(26) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 1) + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 3) + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + (z - 1) + 1 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + (z - 3) + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
g(z, z') -{ 1 }→ 1 + 4 + z' :|: z' >= 0, z = z'

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

g: runtime: O(1) [1], size: O(n1) [5 + z'] f: runtime: ?, size: O(n1) [1 + 6·z]

(27) 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: 11 + 91·z

(28) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 2 }→ g(1 + y + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 0) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
f(z) -{ 3 }→ g(1 + 2 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 3) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + 2
f(z) -{ 3 }→ g(1 + 0 + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + 1) :|: z = 1 + (1 + x' + y') + 0, x' >= 0, y' >= 0
f(z) -{ 3 }→ g(1 + (1 + x'' + y'') + 1, 1 + 0 + g(1 + y'' + f(x''), 1 + x'' + f(y''))) :|: y'' >= 0, z = 1 + 0 + (1 + x'' + y''), x'' >= 0
f(z) -{ 3 }→ g(1 + (1 + x1 + y1) + 3, 1 + 2 + g(1 + y1 + f(x1), 1 + x1 + f(y1))) :|: y1 >= 0, x1 >= 0, z = 1 + 2 + (1 + x1 + y1)
f(z) -{ 3 }→ g(1 + (1 + x2 + y2) + g(1 + y' + f(x'), 1 + x' + f(y')), 1 + (1 + x' + y') + g(1 + y2 + f(x2), 1 + x2 + f(y2))) :|: x' >= 0, y' >= 0, z = 1 + (1 + x' + y') + (1 + x2 + y2), y2 >= 0, x2 >= 0
f(z) -{ 2 }→ g(1 + (1 + x3 + y3) + 0, 1 + x + g(1 + y3 + f(x3), 1 + x3 + f(y3))) :|: z = 1 + x + (1 + x3 + y3), x >= 0, y3 >= 0, x3 >= 0
f(z) -{ 1 }→ 3 :|: z = 2
f(z) -{ 1 }→ 1 :|: z = 0
f(z) -{ 0 }→ 0 :|: z >= 0
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 1 = v0, 1 + 0 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 0 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 1 = v0, 1 + 0 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 1) + 1 = v0, 1 + 0 + 0 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 0, v0 >= 0, v1 >= 0, 1 + 0 + 3 = v0, 1 + 2 + 1 = v1
f(z) -{ 3 }→ 0 :|: z = 1 + 2 + 2, v0 >= 0, v1 >= 0, 1 + 2 + 3 = v0, 1 + 2 + 3 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + (z - 3) + 3 = v0, 1 + 2 + 0 = v1
f(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 + 0 = v0, 1 + (z - 1) + 1 = v1
f(z) -{ 2 }→ 0 :|: z - 3 >= 0, v0 >= 0, v1 >= 0, 1 + 2 + 0 = v0, 1 + (z - 3) + 3 = v1
f(z) -{ 1 }→ 0 :|: z = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, 1 + y + 0 = v0, 1 + x + 0 = v1
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 0, 1 + 0 + 1 = x, x >= 0
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 0 + 2, 1 + 0 + 3 = x, x >= 0, 1 + 2 + 1 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 0, 1 + 2 + 1 = x, x >= 0, 1 + 0 + 3 = x
f(z) -{ 4 }→ 1 + 4 + x :|: z = 1 + 2 + 2, 1 + 2 + 3 = x, x >= 0
f(z) -{ 2 }→ 1 + 4 + x' :|: z = 1 + x + y, x >= 0, y >= 0, 1 + x + 0 = x', x' >= 0, 1 + y + 0 = x'
g(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
g(z, z') -{ 1 }→ 1 + 4 + z' :|: z' >= 0, z = z'

Function symbols to be analyzed:
Previous analysis results are:

g: runtime: O(1) [1], size: O(n1) [5 + z'] f: runtime: O(n1) [11 + 91·z], size: O(n1) [1 + 6·z]

(29) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(30) BOUNDS(1, n^1)