/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRNTS
(19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 578 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 263 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
(32) CpxRNTS
(33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 130 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
(38) CpxRNTS
(39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 758 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 494 ms]
(44) CpxRNTS
(45) FinalProof [FINISHED, 0 ms]
(46) BOUNDS(1, n^2)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(j(x, y), y) -> g(f(x, k(y)))
   f(x, h1(y, z)) -> h2(0, x, h1(y, z))
   g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
   i(f(x, h(y))) -> y
   i(h2(s(x), y, h1(x, z))) -> z
   k(h(x)) -> h1(0, x)
   k(h1(x, y)) -> h1(s(x), y)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) DependencyGraphProof (UPPER BOUND(ID))
The following rules are not reachable from basic terms in the dependency graph and can be removed:

i(f(x, h(y))) -> y
i(h2(s(x), y, h1(x, z))) -> z

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

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


The TRS R consists of the following rules:

   f(j(x, y), y) -> g(f(x, k(y)))
   f(x, h1(y, z)) -> h2(0, x, h1(y, z))
   g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
   k(h(x)) -> h1(0, x)
   k(h1(x, y)) -> h1(s(x), y)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(j(x, y), y) -> g(f(x, k(y)))
   f(x, h1(y, z)) -> h2(0, x, h1(y, z))
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
   k(h(x)) -> h1(0, x)
   k(h1(x, y)) -> h1(s(x), y)
   g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))

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

   h2(x0, x1, x2) -> c_h2(x0, x1, x2)

Rewrite Strategy: FULL
----------------------------------------

(5) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(j(x, y), y) -> g(f(x, k(y)))
   f(x, h1(y, z)) -> h2(0, x, h1(y, z))
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
   k(h(x)) -> h1(0, x)
   k(h1(x, y)) -> h1(s(x), y)
   g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))

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

   h2(x0, x1, x2) -> c_h2(x0, x1, x2)

Rewrite Strategy: INNERMOST
----------------------------------------

(7) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(j(x, y), y) -> g(f(x, k(y))) [1]
   f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1]
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1]
   k(h(x)) -> h1(0, x) [1]
   k(h1(x, y)) -> h1(s(x), y) [1]
   g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1]
   h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(j(x, y), y) -> g(f(x, k(y))) [1]
   f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1]
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1]
   k(h(x)) -> h1(0, x) [1]
   k(h1(x, y)) -> h1(s(x), y) [1]
   g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1]
   h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0]

The TRS has the following type information:
f :: j -> h1:h -> c_h2
j :: j -> h1:h -> j
g :: c_h2 -> c_h2
k :: h1:h -> h1:h
h1 :: 0:s -> a -> h1:h
h2 :: 0:s -> j -> h1:h -> c_h2
0 :: 0:s
s :: 0:s -> 0:s
h :: a -> h1:h
c_h2 :: 0:s -> j -> h1:h -> c_h2

Rewrite Strategy: INNERMOST
----------------------------------------

(11) 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:
none

(c) The following functions are completely defined:

   f_2
   k_1
   g_1
   h2_3

Due to the following rules being added:

   h2(v0, v1, v2) -> const [0]
   f(v0, v1) -> const [0]
   k(v0) -> const2 [0]
   g(v0) -> const [0]

And the following fresh constants:    const, const2, const1, const3

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(j(x, y), y) -> g(f(x, k(y))) [1]
   f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1]
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1]
   k(h(x)) -> h1(0, x) [1]
   k(h1(x, y)) -> h1(s(x), y) [1]
   g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1]
   h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0]
   h2(v0, v1, v2) -> const [0]
   f(v0, v1) -> const [0]
   k(v0) -> const2 [0]
   g(v0) -> const [0]

The TRS has the following type information:
f :: j -> h1:h:const2 -> c_h2:const
j :: j -> h1:h:const2 -> j
g :: c_h2:const -> c_h2:const
k :: h1:h:const2 -> h1:h:const2
h1 :: 0:s -> a -> h1:h:const2
h2 :: 0:s -> j -> h1:h:const2 -> c_h2:const
0 :: 0:s
s :: 0:s -> 0:s
h :: a -> h1:h:const2
c_h2 :: 0:s -> j -> h1:h:const2 -> c_h2:const
const :: c_h2:const
const2 :: h1:h:const2
const1 :: j
const3 :: a

Rewrite Strategy: INNERMOST
----------------------------------------

(13) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(j(x, h(x')), h(x')) -> g(f(x, h1(0, x'))) [2]
   f(j(x, h1(x'', y')), h1(x'', y')) -> g(f(x, h1(s(x''), y'))) [2]
   f(j(x, y), y) -> g(f(x, const2)) [1]
   f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1]
   h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1]
   k(h(x)) -> h1(0, x) [1]
   k(h1(x, y)) -> h1(s(x), y) [1]
   g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1]
   h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0]
   h2(v0, v1, v2) -> const [0]
   f(v0, v1) -> const [0]
   k(v0) -> const2 [0]
   g(v0) -> const [0]

The TRS has the following type information:
f :: j -> h1:h:const2 -> c_h2:const
j :: j -> h1:h:const2 -> j
g :: c_h2:const -> c_h2:const
k :: h1:h:const2 -> h1:h:const2
h1 :: 0:s -> a -> h1:h:const2
h2 :: 0:s -> j -> h1:h:const2 -> c_h2:const
0 :: 0:s
s :: 0:s -> 0:s
h :: a -> h1:h:const2
c_h2 :: 0:s -> j -> h1:h:const2 -> c_h2:const
const :: c_h2:const
const2 :: h1:h:const2
const1 :: j
const3 :: a

Rewrite Strategy: INNERMOST
----------------------------------------

(15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   const => 0
   const2 => 0
   const1 => 0
   const3 => 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z', z'') -{ 1 }-> h2(0, x, 1 + y + z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z'' = y, z' = 1 + x + y, x >= 0, y >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + x')) :|: z' = 1 + x + (1 + x'), z'' = 1 + x', x >= 0, x' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   h2(z', z'', z1) -{ 1 }-> h2(1 + x, y, 1 + (1 + z) + u) :|: z >= 0, z' = x, x >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   h2(z', z'', z1) -{ 0 }-> 1 + x0 + x1 + x2 :|: z'' = x1, x0 >= 0, x1 >= 0, z1 = x2, x2 >= 0, z' = x0
   k(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   k(z') -{ 1 }-> 1 + 0 + x :|: z' = 1 + x, x >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0


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

(17) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0


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

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

   { h2 }
   { k }
   { g }
   { f }

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

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

   f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {h2}, {k}, {g}, {f}

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

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {h2}, {k}, {g}, {f}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: h2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z' + z'' + z1

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

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

   f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {h2}, {k}, {g}, {f}
Previous analysis results are:
  h2: runtime: ?, size: O(n^1) [1 + z' + z'' + z1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: h2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z''

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

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

   f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {k}, {g}, {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {k}, {g}, {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {k}, {g}, {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: ?, size: O(n^1) [1 + z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: O(1) [1], size: O(n^1) [1 + z']

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: O(1) [1], size: O(n^1) [1 + z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: O(1) [1], size: O(n^1) [1 + z']
  g: runtime: ?, size: O(n^1) [1 + z']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: O(1) [1], size: O(n^1) [1 + z']
  g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: O(1) [1], size: O(n^1) [1 + z']
  g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: O(1) [1], size: O(n^1) [1 + z']
  g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']
  f: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2

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

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

   f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
   f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0
   f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0
   f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0
   g(z') -{ 0 }-> 0 :|: z' >= 0
   h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0
   h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0
   k(z') -{ 0 }-> 0 :|: z' >= 0
   k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0
   k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1]
  k: runtime: O(1) [1], size: O(n^1) [1 + z']
  g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']
  f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z'']

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

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

(46)
BOUNDS(1, n^2)