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

(0) Obligation:

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


The TRS R consists of the following rules:

lt0(Nil, Cons(x', xs)) → True
lt0(Cons(x', xs'), Cons(x, xs)) → lt0(xs', xs)
g(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lt0(x, Nil) → False
g(x, Cons(x', xs)) → g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
f(x, Cons(x', xs)) → f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
number4(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(x, y) → Cons(f(x, y), Cons(g(x, y), Nil))

The (relative) TRS S consists of the following rules:

g[Ite][False][Ite](False, Cons(x, xs), y) → g(xs, Cons(Cons(Nil, Nil), y))
g[Ite][False][Ite](True, x', Cons(x, xs)) → g(x', xs)
f[Ite][False][Ite](False, Cons(x, xs), y) → f(xs, Cons(Cons(Nil, Nil), y))
f[Ite][False][Ite](True, x', Cons(x, xs)) → f(x', xs)

Rewrite Strategy: INNERMOST

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

Transformed relative 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:

lt0(Nil, Cons(x', xs)) → True [1]
lt0(Cons(x', xs'), Cons(x, xs)) → lt0(xs', xs) [1]
g(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1]
f(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
lt0(x, Nil) → False [1]
g(x, Cons(x', xs)) → g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1]
f(x, Cons(x', xs)) → f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1]
number4(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1]
goal(x, y) → Cons(f(x, y), Cons(g(x, y), Nil)) [1]
g[Ite][False][Ite](False, Cons(x, xs), y) → g(xs, Cons(Cons(Nil, Nil), y)) [0]
g[Ite][False][Ite](True, x', Cons(x, xs)) → g(x', xs) [0]
f[Ite][False][Ite](False, Cons(x, xs), y) → f(xs, Cons(Cons(Nil, Nil), y)) [0]
f[Ite][False][Ite](True, x', Cons(x, xs)) → f(x', xs) [0]

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:

lt0(Nil, Cons(x', xs)) → True [1]
lt0(Cons(x', xs'), Cons(x, xs)) → lt0(xs', xs) [1]
g(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1]
f(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
lt0(x, Nil) → False [1]
g(x, Cons(x', xs)) → g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1]
f(x, Cons(x', xs)) → f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1]
number4(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1]
goal(x, y) → Cons(f(x, y), Cons(g(x, y), Nil)) [1]
g[Ite][False][Ite](False, Cons(x, xs), y) → g(xs, Cons(Cons(Nil, Nil), y)) [0]
g[Ite][False][Ite](True, x', Cons(x, xs)) → g(x', xs) [0]
f[Ite][False][Ite](False, Cons(x, xs), y) → f(xs, Cons(Cons(Nil, Nil), y)) [0]
f[Ite][False][Ite](True, x', Cons(x, xs)) → f(x', xs) [0]

The TRS has the following type information:

lt0 :: Nil:Cons → Nil:Cons → True:False Nil :: Nil:Cons Cons :: Nil:Cons → Nil:Cons → Nil:Cons True :: True:False g :: Nil:Cons → Nil:Cons → Nil:Cons f :: Nil:Cons → Nil:Cons → Nil:Cons notEmpty :: Nil:Cons → True:False False :: True:False g[Ite][False][Ite] :: True:False → Nil:Cons → Nil:Cons → Nil:Cons f[Ite][False][Ite] :: True:False → Nil:Cons → Nil:Cons → Nil:Cons number4 :: a → Nil:Cons goal :: Nil:Cons → Nil:Cons → Nil: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:

g
f
notEmpty
number4
goal

(c) The following functions are completely defined:

lt0
g[Ite][False][Ite]
f[Ite][False][Ite]

Due to the following rules being added:

g[Ite][False][Ite](v0, v1, v2) → Nil [0]
f[Ite][False][Ite](v0, v1, v2) → Nil [0]

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: lt0(Nil, Cons(x', xs)) → True [1] lt0(Cons(x', xs'), Cons(x, xs)) → lt0(xs', xs) [1] g(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] f(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] lt0(x, Nil) → False [1] g(x, Cons(x', xs)) → g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] f(x, Cons(x', xs)) → f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)) [1] number4(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] goal(x, y) → Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) → g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) → g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) → f(xs, Cons(Cons(Nil, Nil), y)) [0] f[Ite][False][Ite](True, x', Cons(x, xs)) → f(x', xs) [0] g[Ite][False][Ite](v0, v1, v2) → Nil [0] f[Ite][False][Ite](v0, v1, v2) → Nil [0] The TRS has the following type information: lt0 :: Nil:Cons → Nil:Cons → True:False Nil :: Nil:Cons Cons :: Nil:Cons → Nil:Cons → Nil:Cons True :: True:False g :: Nil:Cons → Nil:Cons → Nil:Cons f :: Nil:Cons → Nil:Cons → Nil:Cons notEmpty :: Nil:Cons → True:False False :: True:False g[Ite][False][Ite] :: True:False → Nil:Cons → Nil:Cons → Nil:Cons f[Ite][False][Ite] :: True:False → Nil:Cons → Nil:Cons → Nil:Cons number4 :: a → Nil:Cons goal :: Nil:Cons → Nil:Cons → Nil: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: lt0(Nil, Cons(x', xs)) → True [1] lt0(Cons(x', xs'), Cons(x, xs)) → lt0(xs', xs) [1] g(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] f(x, Nil) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] lt0(x, Nil) → False [1] g(Nil, Cons(x', xs)) → g[Ite][False][Ite](True, Nil, Cons(x', xs)) [2] g(Cons(x'', xs''), Cons(x', xs)) → g[Ite][False][Ite](lt0(xs'', Nil), Cons(x'', xs''), Cons(x', xs)) [2] f(Nil, Cons(x', xs)) → f[Ite][False][Ite](True, Nil, Cons(x', xs)) [2] f(Cons(x''', xs'''), Cons(x', xs)) → f[Ite][False][Ite](lt0(xs''', Nil), Cons(x''', xs'''), Cons(x', xs)) [2] number4(n) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))) [1] goal(x, y) → Cons(f(x, y), Cons(g(x, y), Nil)) [1] g[Ite][False][Ite](False, Cons(x, xs), y) → g(xs, Cons(Cons(Nil, Nil), y)) [0] g[Ite][False][Ite](True, x', Cons(x, xs)) → g(x', xs) [0] f[Ite][False][Ite](False, Cons(x, xs), y) → f(xs, Cons(Cons(Nil, Nil), y)) [0] f[Ite][False][Ite](True, x', Cons(x, xs)) → f(x', xs) [0] g[Ite][False][Ite](v0, v1, v2) → Nil [0] f[Ite][False][Ite](v0, v1, v2) → Nil [0] The TRS has the following type information: lt0 :: Nil:Cons → Nil:Cons → True:False Nil :: Nil:Cons Cons :: Nil:Cons → Nil:Cons → Nil:Cons True :: True:False g :: Nil:Cons → Nil:Cons → Nil:Cons f :: Nil:Cons → Nil:Cons → Nil:Cons notEmpty :: Nil:Cons → True:False False :: True:False g[Ite][False][Ite] :: True:False → Nil:Cons → Nil:Cons → Nil:Cons f[Ite][False][Ite] :: True:False → Nil:Cons → Nil:Cons → Nil:Cons number4 :: a → Nil:Cons goal :: Nil:Cons → Nil:Cons → Nil: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:

Nil => 0
True => 1
False => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 2 }→ f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: x >= 0, z = x, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + y) :|: xs >= 0, z' = 1 + x + xs, z'' = y, x >= 0, y >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: x >= 0, z = x, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + y) :|: xs >= 0, z' = 1 + x + xs, z'' = y, x >= 0, y >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
goal(z, z') -{ 1 }→ 1 + f(x, y) + (1 + g(x, y) + 0) :|: x >= 0, y >= 0, z = x, z' = y
lt0(z, z') -{ 1 }→ lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: n >= 0, z = n

(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') -{ 2 }→ f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 1 }→ lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ lt0 } { notEmpty } { number4 } { g[Ite][False][Ite], g } { f[Ite][False][Ite], f } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 2 }→ f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 1 }→ lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {lt0}, {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}

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

Computed SIZE bound using CoFloCo for: lt0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(16) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 2 }→ f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 1 }→ lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {lt0}, {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using PUBS for: lt0
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') -{ 2 }→ f[Ite][False][Ite](lt0(xs''', 0), 1 + x''' + xs''', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](lt0(xs'', 0), 1 + x'' + xs'', 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 1 }→ lt0(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1]

(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') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1]

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

Computed SIZE bound using CoFloCo for: notEmpty
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, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {notEmpty}, {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using CoFloCo for: notEmpty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(24) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: number4
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 4

(28) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {number4}, {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: ?, size: O(1) [4]

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

Computed RUNTIME bound using CoFloCo for: number4
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(30) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4]

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4]

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: g[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 4

Computed SIZE bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 4

(34) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {g[Ite][False][Ite],g}, {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: ?, size: O(1) [4] g: runtime: ?, size: O(1) [4]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: g[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 4 + 9·z' + 3·z''

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 7 + 9·z + 3·z'

(36) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 3 }→ g[Ite][False][Ite](s', 1 + x'' + xs'', 1 + x' + xs) :|: s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 2 }→ g[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 0 }→ g(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 1 }→ 1 + f(z, z') + (1 + g(z, z') + 0) :|: z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] g: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4]

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 9 + 3·x' + 3·xs }→ s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 19 + 3·x' + 9·x'' + 3·xs + 9·xs'' }→ s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 8 + 9·z + 3·z' }→ 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] g: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: f[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 4

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 4

(40) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 9 + 3·x' + 3·xs }→ s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 19 + 3·x' + 9·x'' + 3·xs + 9·xs'' }→ s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 8 + 9·z + 3·z' }→ 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed: {f[Ite][False][Ite],f}, {goal}
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] g: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4] f[Ite][False][Ite]: runtime: ?, size: O(1) [4] f: runtime: ?, size: O(1) [4]

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: f[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 4 + 9·z' + 3·z''

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 7 + 9·z + 3·z'

(42) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 3 }→ f[Ite][False][Ite](s'', 1 + x''' + xs''', 1 + x' + xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 2 }→ f[Ite][False][Ite](1, 0, 1 + x' + xs) :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(xs, 1 + (1 + 0 + 0) + z'') :|: xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ f(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 9 + 3·x' + 3·xs }→ s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 19 + 3·x' + 9·x'' + 3·xs + 9·xs'' }→ s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 8 + 9·z + 3·z' }→ 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

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

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] g: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] f: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4]

(43) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(44) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 9 + 3·x' + 3·xs }→ s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 19 + 3·x' + 9·x''' + 3·xs + 9·xs''' }→ s7 :|: s7 >= 0, s7 <= 4, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s10 :|: s10 >= 0, s10 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s9 :|: s9 >= 0, s9 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 9 + 3·x' + 3·xs }→ s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 19 + 3·x' + 9·x'' + 3·xs + 9·xs'' }→ s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 15 + 18·z + 6·z' }→ 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

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

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] g: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] f: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4]

(45) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 10

(46) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 9 + 3·x' + 3·xs }→ s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 19 + 3·x' + 9·x''' + 3·xs + 9·xs''' }→ s7 :|: s7 >= 0, s7 <= 4, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s10 :|: s10 >= 0, s10 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s9 :|: s9 >= 0, s9 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 9 + 3·x' + 3·xs }→ s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 19 + 3·x' + 9·x'' + 3·xs + 9·xs'' }→ s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 15 + 18·z + 6·z' }→ 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

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

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] g: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] f: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4] goal: runtime: ?, size: O(1) [10]

(47) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 15 + 18·z + 6·z'

(48) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 9 + 3·x' + 3·xs }→ s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
f(z, z') -{ 19 + 3·x' + 9·x''' + 3·xs + 9·xs''' }→ s7 :|: s7 >= 0, s7 <= 4, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + x' + xs, x' >= 0, xs''' >= 0, x''' >= 0, z = 1 + x''' + xs'''
f(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
f[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s10 :|: s10 >= 0, s10 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
f[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s9 :|: s9 >= 0, s9 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
f[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
g(z, z') -{ 9 + 3·x' + 3·xs }→ s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
g(z, z') -{ 19 + 3·x' + 9·x'' + 3·xs + 9·xs'' }→ s2 :|: s2 >= 0, s2 <= 4, s' >= 0, s' <= 1, xs >= 0, z' = 1 + x' + xs, xs'' >= 0, x' >= 0, z = 1 + x'' + xs'', x'' >= 0
g(z, z') -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0, z' = 0
g[Ite][False][Ite](z, z', z'') -{ 13 + 9·xs + 3·z'' }→ s4 :|: s4 >= 0, s4 <= 4, xs >= 0, z' = 1 + x + xs, x >= 0, z'' >= 0, z = 0
g[Ite][False][Ite](z, z', z'') -{ 7 + 3·xs + 9·z' }→ s5 :|: s5 >= 0, s5 <= 4, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
g[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
goal(z, z') -{ 15 + 18·z + 6·z' }→ 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0
lt0(z, z') -{ 2 + xs }→ s :|: s >= 0, s <= 1, xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
lt0(z, z') -{ 1 }→ 1 :|: xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
lt0(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
number4(z) -{ 1 }→ 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))) :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:

lt0: runtime: O(n1) [1 + z'], size: O(1) [1] notEmpty: runtime: O(1) [1], size: O(1) [1] number4: runtime: O(1) [1], size: O(1) [4] g[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] g: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4] f[Ite][False][Ite]: runtime: O(n1) [4 + 9·z' + 3·z''], size: O(1) [4] f: runtime: O(n1) [7 + 9·z + 3·z'], size: O(1) [4] goal: runtime: O(n1) [15 + 18·z + 6·z'], size: O(1) [10]

(49) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(50) BOUNDS(1, n^1)