WORST_CASE(?,O(n^1))
proof of input_wRrsFwXnXs.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 209 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxWeightedTrs
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTypedWeightedTrs
(7) CompletionProof [UPPER BOUND(ID), 4 ms]
(8) CpxTypedWeightedCompleteTrs
(9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedCompleteTrs
(11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(12) CpxRNTS
(13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 285 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 133 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 135 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 111 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
(34) CpxRNTS
(35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 1599 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 719 ms]
(40) CpxRNTS
(41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 1551 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 676 ms]
(46) CpxRNTS
(47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(48) CpxRNTS
(49) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
(50) CpxRNTS
(51) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
(52) CpxRNTS
(53) FinalProof [FINISHED, 0 ms]
(54) BOUNDS(1, n^1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative 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:

   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
----------------------------------------

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
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
----------------------------------------

(7) CompletionProof (UPPER BOUND(ID))
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_2
   f_2
   notEmpty_1
   number4_1
   goal_2

(c) The following functions are completely defined:

   lt0_2
   g[Ite][False][Ite]_3
   f[Ite][False][Ite]_3

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

----------------------------------------

(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(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
----------------------------------------

(9) NarrowingProof (BOTH BOUNDS(ID, ID))
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:

   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
----------------------------------------

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
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

----------------------------------------

(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))) :|: 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


----------------------------------------

(13) SimplificationProof (BOTH BOUNDS(ID, ID))
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, 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


----------------------------------------

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

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

----------------------------------------

(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}

----------------------------------------

(17) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

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

----------------------------------------

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, 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]

----------------------------------------

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: lt0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

----------------------------------------

(22)
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(n^1) [2 + z'], size: O(1) [1]

----------------------------------------

(23) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + z'], size: O(1) [1]

----------------------------------------

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + z'], size: O(1) [1]
  notEmpty: runtime: ?, size: O(1) [1]

----------------------------------------

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + z'], size: O(1) [1]
  notEmpty: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(29) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + z'], size: O(1) [1]
  notEmpty: runtime: O(1) [1], size: O(1) [1]

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + z'], size: O(1) [1]
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  number4: runtime: ?, size: O(1) [4]

----------------------------------------

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + z'], size: O(1) [1]
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  number4: runtime: O(1) [1], size: O(1) [4]

----------------------------------------

(35) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + z'], size: O(1) [1]
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  number4: runtime: O(1) [1], size: O(1) [4]

----------------------------------------

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + 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]

----------------------------------------

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: g[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 19 + 12*z' + 4*z''

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 23 + 12*z + 4*z'

----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 4 }-> 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') -{ 3 + 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(n^1) [2 + 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(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]

----------------------------------------

(41) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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'') -{ 23 + 4*xs + 12*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') -{ 24 + 12*z + 4*z' }-> 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0
   lt0(z, z') -{ 3 + 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(n^1) [2 + 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(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]

----------------------------------------

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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'') -{ 23 + 4*xs + 12*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') -{ 24 + 12*z + 4*z' }-> 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0
   lt0(z, z') -{ 3 + 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(n^1) [2 + 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(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]
  f[Ite][False][Ite]: runtime: ?, size: O(1) [4]
  f: runtime: ?, size: O(1) [4]

----------------------------------------

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 19 + 12*z' + 4*z''

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 23 + 12*z + 4*z'

----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> 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') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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'') -{ 23 + 4*xs + 12*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') -{ 24 + 12*z + 4*z' }-> 1 + f(z, z') + (1 + s3 + 0) :|: s3 >= 0, s3 <= 4, z >= 0, z' >= 0
   lt0(z, z') -{ 3 + 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(n^1) [2 + 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(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]
  f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]

----------------------------------------

(47) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 25 + 4*x' + 4*xs }-> s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   f(z, z') -{ 39 + 4*x' + 12*x''' + 4*xs + 12*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'') -{ 23 + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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'') -{ 23 + 4*xs + 12*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') -{ 47 + 24*z + 8*z' }-> 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0
   lt0(z, z') -{ 3 + 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(n^1) [2 + 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(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]
  f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]

----------------------------------------

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 25 + 4*x' + 4*xs }-> s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   f(z, z') -{ 39 + 4*x' + 12*x''' + 4*xs + 12*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'') -{ 23 + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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'') -{ 23 + 4*xs + 12*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') -{ 47 + 24*z + 8*z' }-> 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0
   lt0(z, z') -{ 3 + 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(n^1) [2 + 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(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]
  f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]
  goal: runtime: ?, size: O(1) [10]

----------------------------------------

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 47 + 24*z + 8*z'

----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 25 + 4*x' + 4*xs }-> s6 :|: s6 >= 0, s6 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   f(z, z') -{ 39 + 4*x' + 12*x''' + 4*xs + 12*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'') -{ 23 + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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') -{ 25 + 4*x' + 4*xs }-> s1 :|: s1 >= 0, s1 <= 4, xs >= 0, z' = 1 + x' + xs, x' >= 0, z = 0
   g(z, z') -{ 39 + 4*x' + 12*x'' + 4*xs + 12*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'') -{ 31 + 12*xs + 4*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'') -{ 23 + 4*xs + 12*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') -{ 47 + 24*z + 8*z' }-> 1 + s8 + (1 + s3 + 0) :|: s8 >= 0, s8 <= 4, s3 >= 0, s3 <= 4, z >= 0, z' >= 0
   lt0(z, z') -{ 3 + 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(n^1) [2 + 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(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  g: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]
  f[Ite][False][Ite]: runtime: O(n^1) [19 + 12*z' + 4*z''], size: O(1) [4]
  f: runtime: O(n^1) [23 + 12*z + 4*z'], size: O(1) [4]
  goal: runtime: O(n^1) [47 + 24*z + 8*z'], size: O(1) [10]

----------------------------------------

(53) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(54)
BOUNDS(1, n^1)