WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 181 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(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), 250 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 445 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 125 ms]
(32) CpxRNTS
(33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 275 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 337 ms]
(38) CpxRNTS
(39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 211 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 232 ms]
(48) CpxRNTS
(49) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
(50) CpxRNTS
(51) FinalProof [FINISHED, 0 ms]
(52) BOUNDS(1, n^2)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(g(x), x, y) -> f(y, y, g(y))
   g(g(x)) -> g(x)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))


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

(2)
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(g(x), x, y) -> f(y, y, g(y))
   g(g(x)) -> g(x)

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

   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))

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

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

(4)
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(g(x), x, y) -> f(y, y, g(y))
   g(g(x)) -> g(x)

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

   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))

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

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

(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(c_g(x), x, y) -> f(y, y, g(y))
   g(c_g(x)) -> g(x)

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

   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))
   g(x0) -> c_g(x0)

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(c_g(x), x, y) -> f(y, y, g(y)) [1]
   g(c_g(x)) -> g(x) [1]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   g(x0) -> c_g(x0) [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(c_g(x), x, y) -> f(y, y, g(y)) [1]
   g(c_g(x)) -> g(x) [1]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   g(x0) -> c_g(x0) [0]

The TRS has the following type information:
f :: c_g -> c_g -> c_g -> c_g
c_g :: c_g -> c_g
g :: c_g -> c_g
encArg :: cons_f:cons_g -> c_g
cons_f :: cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g
cons_g :: cons_f:cons_g -> cons_f:cons_g
encode_f :: cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g -> c_g
encode_g :: cons_f:cons_g -> c_g

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_3
   encArg_1
   encode_f_3
   encode_g_1
   g_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_f(v0, v1, v2) -> const [0]
   encode_g(v0) -> const [0]
   g(v0) -> const [0]
   f(v0, v1, v2) -> const [0]

And the following fresh constants:    const, const1

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

(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(c_g(x), x, y) -> f(y, y, g(y)) [1]
   g(c_g(x)) -> g(x) [1]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   g(x0) -> c_g(x0) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1, v2) -> const [0]
   encode_g(v0) -> const [0]
   g(v0) -> const [0]
   f(v0, v1, v2) -> const [0]

The TRS has the following type information:
f :: c_g:const -> c_g:const -> c_g:const -> c_g:const
c_g :: c_g:const -> c_g:const
g :: c_g:const -> c_g:const
encArg :: cons_f:cons_g -> c_g:const
cons_f :: cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g
cons_g :: cons_f:cons_g -> cons_f:cons_g
encode_f :: cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g -> c_g:const
encode_g :: cons_f:cons_g -> c_g:const
const :: c_g:const
const1 :: cons_f:cons_g

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(c_g(x), x, c_g(x')) -> f(c_g(x'), c_g(x'), g(x')) [2]
   f(c_g(x), x, y) -> f(y, y, c_g(y)) [1]
   f(c_g(x), x, y) -> f(y, y, const) [1]
   g(c_g(x)) -> g(x) [1]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(cons_f(x_125, x_212, x_312))) -> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) [0]
   encArg(cons_g(cons_g(x_126))) -> g(g(encArg(x_126))) [0]
   encArg(cons_g(x_1)) -> g(const) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(cons_f(x_153, x_226, x_326)) -> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) [0]
   encode_g(cons_g(x_154)) -> g(g(encArg(x_154))) [0]
   encode_g(x_1) -> g(const) [0]
   g(x0) -> c_g(x0) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1, v2) -> const [0]
   encode_g(v0) -> const [0]
   g(v0) -> const [0]
   f(v0, v1, v2) -> const [0]

The TRS has the following type information:
f :: c_g:const -> c_g:const -> c_g:const -> c_g:const
c_g :: c_g:const -> c_g:const
g :: c_g:const -> c_g:const
encArg :: cons_f:cons_g -> c_g:const
cons_f :: cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g
cons_g :: cons_f:cons_g -> cons_f:cons_g
encode_f :: cons_f:cons_g -> cons_f:cons_g -> cons_f:cons_g -> c_g:const
encode_g :: cons_f:cons_g -> c_g:const
const :: c_g:const
const1 :: cons_f:cons_g

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:

   const => 0
   const1 => 0

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

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

   encArg(z) -{ 0 }-> g(g(encArg(x_126))) :|: z = 1 + (1 + x_126), x_126 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> g(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(x_154))) :|: x_154 >= 0, z = 1 + x_154
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> g(0) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z, z', z'') -{ 1 }-> f(y, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 1 + x
   f(z, z', z'') -{ 1 }-> f(y, y, 1 + y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 1 + x
   f(z, z', z'') -{ 2 }-> f(1 + x', 1 + x', g(x')) :|: z' = x, z'' = 1 + x', x >= 0, x' >= 0, z = 1 + x
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   g(z) -{ 1 }-> g(x) :|: x >= 0, z = 1 + x
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 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:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> g(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> g(0) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + (z'' - 1), g(z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ 1 }-> g(z - 1) :|: z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0


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

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

   { g }
   { f }
   { encArg }
   { encode_f }
   { encode_g }

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

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

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> g(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> g(0) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + (z'' - 1), g(z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ 1 }-> g(z - 1) :|: z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_f}, {encode_g}

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

(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:

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> g(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> g(0) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + (z'' - 1), g(z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ 1 }-> g(z - 1) :|: z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_f}, {encode_g}

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

(23) 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

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

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

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> g(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> g(0) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + (z'' - 1), g(z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ 1 }-> g(z - 1) :|: z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

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

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

(25) 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: z

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

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

   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> g(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> g(0) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> f(1 + (z'' - 1), 1 + (z'' - 1), g(z'' - 1)) :|: z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ 1 }-> g(z - 1) :|: z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]

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

(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:

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 + z'' }-> f(1 + (z'' - 1), 1 + (z'' - 1), s) :|: s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 + z'' }-> f(1 + (z'' - 1), 1 + (z'' - 1), s) :|: s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: ?, size: O(1) [0]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 1 }-> f(z'', z'', 0) :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 }-> f(z'', z'', 1 + z'') :|: z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 1 + z'' }-> f(1 + (z'' - 1), 1 + (z'' - 1), s) :|: s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]

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

(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:

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 3*z + 2*z^2

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> g(g(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> g(f(encArg(x_125), encArg(x_212), encArg(x_312))) :|: x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 0 }-> g(g(encArg(z - 1))) :|: z - 1 >= 0
   encode_g(z) -{ 0 }-> g(f(encArg(x_153), encArg(x_226), encArg(x_326))) :|: x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: O(n^2) [3*z + 2*z^2], size: O(n^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:

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1 + s11 + s12 + 3*x_125 + 2*x_125^2 + 3*x_212 + 2*x_212^2 + 3*x_312 + 2*x_312^2 }-> s13 :|: s9 >= 0, s9 <= x_125, s10 >= 0, s10 <= x_212, s11 >= 0, s11 <= x_312, s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12 + 1, x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 2 + s14 + s15 + -5*z + 2*z^2 }-> s16 :|: s14 >= 0, s14 <= z - 2, s15 >= 0, s15 <= s14 + 1, s16 >= 0, s16 <= s15 + 1, z - 2 >= 0
   encArg(z) -{ 1 + s7 + 3*x_1 + 2*x_1^2 + 3*x_2 + 2*x_2^2 + 3*x_3 + 2*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 1 + s19 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 3*z'' + 2*z''^2 }-> s20 :|: s17 >= 0, s17 <= z, s18 >= 0, s18 <= z', s19 >= 0, s19 <= z'', s20 >= 0, s20 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 1 + s23 + s24 + 3*x_153 + 2*x_153^2 + 3*x_226 + 2*x_226^2 + 3*x_326 + 2*x_326^2 }-> s25 :|: s21 >= 0, s21 <= x_153, s22 >= 0, s22 <= x_226, s23 >= 0, s23 <= x_326, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= s24 + 1, x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ -1 + s26 + s27 + -1*z + 2*z^2 }-> s28 :|: s26 >= 0, s26 <= z - 1, s27 >= 0, s27 <= s26 + 1, s28 >= 0, s28 <= s27 + 1, z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: O(n^2) [3*z + 2*z^2], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1 + s11 + s12 + 3*x_125 + 2*x_125^2 + 3*x_212 + 2*x_212^2 + 3*x_312 + 2*x_312^2 }-> s13 :|: s9 >= 0, s9 <= x_125, s10 >= 0, s10 <= x_212, s11 >= 0, s11 <= x_312, s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12 + 1, x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 2 + s14 + s15 + -5*z + 2*z^2 }-> s16 :|: s14 >= 0, s14 <= z - 2, s15 >= 0, s15 <= s14 + 1, s16 >= 0, s16 <= s15 + 1, z - 2 >= 0
   encArg(z) -{ 1 + s7 + 3*x_1 + 2*x_1^2 + 3*x_2 + 2*x_2^2 + 3*x_3 + 2*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 1 + s19 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 3*z'' + 2*z''^2 }-> s20 :|: s17 >= 0, s17 <= z, s18 >= 0, s18 <= z', s19 >= 0, s19 <= z'', s20 >= 0, s20 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 1 + s23 + s24 + 3*x_153 + 2*x_153^2 + 3*x_226 + 2*x_226^2 + 3*x_326 + 2*x_326^2 }-> s25 :|: s21 >= 0, s21 <= x_153, s22 >= 0, s22 <= x_226, s23 >= 0, s23 <= x_326, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= s24 + 1, x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ -1 + s26 + s27 + -1*z + 2*z^2 }-> s28 :|: s26 >= 0, s26 <= z - 1, s27 >= 0, s27 <= s26 + 1, s28 >= 0, s28 <= s27 + 1, z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: O(n^2) [3*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: ?, size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 4*z'' + 2*z''^2

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1 + s11 + s12 + 3*x_125 + 2*x_125^2 + 3*x_212 + 2*x_212^2 + 3*x_312 + 2*x_312^2 }-> s13 :|: s9 >= 0, s9 <= x_125, s10 >= 0, s10 <= x_212, s11 >= 0, s11 <= x_312, s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12 + 1, x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 2 + s14 + s15 + -5*z + 2*z^2 }-> s16 :|: s14 >= 0, s14 <= z - 2, s15 >= 0, s15 <= s14 + 1, s16 >= 0, s16 <= s15 + 1, z - 2 >= 0
   encArg(z) -{ 1 + s7 + 3*x_1 + 2*x_1^2 + 3*x_2 + 2*x_2^2 + 3*x_3 + 2*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 1 + s19 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 3*z'' + 2*z''^2 }-> s20 :|: s17 >= 0, s17 <= z, s18 >= 0, s18 <= z', s19 >= 0, s19 <= z'', s20 >= 0, s20 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 1 + s23 + s24 + 3*x_153 + 2*x_153^2 + 3*x_226 + 2*x_226^2 + 3*x_326 + 2*x_326^2 }-> s25 :|: s21 >= 0, s21 <= x_153, s22 >= 0, s22 <= x_226, s23 >= 0, s23 <= x_326, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= s24 + 1, x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ -1 + s26 + s27 + -1*z + 2*z^2 }-> s28 :|: s26 >= 0, s26 <= z - 1, s27 >= 0, s27 <= s26 + 1, s28 >= 0, s28 <= s27 + 1, z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: O(n^2) [3*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [1 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 4*z'' + 2*z''^2], size: O(1) [0]

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

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1 + s11 + s12 + 3*x_125 + 2*x_125^2 + 3*x_212 + 2*x_212^2 + 3*x_312 + 2*x_312^2 }-> s13 :|: s9 >= 0, s9 <= x_125, s10 >= 0, s10 <= x_212, s11 >= 0, s11 <= x_312, s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12 + 1, x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 2 + s14 + s15 + -5*z + 2*z^2 }-> s16 :|: s14 >= 0, s14 <= z - 2, s15 >= 0, s15 <= s14 + 1, s16 >= 0, s16 <= s15 + 1, z - 2 >= 0
   encArg(z) -{ 1 + s7 + 3*x_1 + 2*x_1^2 + 3*x_2 + 2*x_2^2 + 3*x_3 + 2*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 1 + s19 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 3*z'' + 2*z''^2 }-> s20 :|: s17 >= 0, s17 <= z, s18 >= 0, s18 <= z', s19 >= 0, s19 <= z'', s20 >= 0, s20 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 1 + s23 + s24 + 3*x_153 + 2*x_153^2 + 3*x_226 + 2*x_226^2 + 3*x_326 + 2*x_326^2 }-> s25 :|: s21 >= 0, s21 <= x_153, s22 >= 0, s22 <= x_226, s23 >= 0, s23 <= x_326, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= s24 + 1, x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ -1 + s26 + s27 + -1*z + 2*z^2 }-> s28 :|: s26 >= 0, s26 <= z - 1, s27 >= 0, s27 <= s26 + 1, s28 >= 0, s28 <= s27 + 1, z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: O(n^2) [3*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [1 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 4*z'' + 2*z''^2], size: O(1) [0]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1 + s11 + s12 + 3*x_125 + 2*x_125^2 + 3*x_212 + 2*x_212^2 + 3*x_312 + 2*x_312^2 }-> s13 :|: s9 >= 0, s9 <= x_125, s10 >= 0, s10 <= x_212, s11 >= 0, s11 <= x_312, s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12 + 1, x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 2 + s14 + s15 + -5*z + 2*z^2 }-> s16 :|: s14 >= 0, s14 <= z - 2, s15 >= 0, s15 <= s14 + 1, s16 >= 0, s16 <= s15 + 1, z - 2 >= 0
   encArg(z) -{ 1 + s7 + 3*x_1 + 2*x_1^2 + 3*x_2 + 2*x_2^2 + 3*x_3 + 2*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 1 + s19 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 3*z'' + 2*z''^2 }-> s20 :|: s17 >= 0, s17 <= z, s18 >= 0, s18 <= z', s19 >= 0, s19 <= z'', s20 >= 0, s20 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 1 + s23 + s24 + 3*x_153 + 2*x_153^2 + 3*x_226 + 2*x_226^2 + 3*x_326 + 2*x_326^2 }-> s25 :|: s21 >= 0, s21 <= x_153, s22 >= 0, s22 <= x_226, s23 >= 0, s23 <= x_326, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= s24 + 1, x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ -1 + s26 + s27 + -1*z + 2*z^2 }-> s28 :|: s26 >= 0, s26 <= z - 1, s27 >= 0, s27 <= s26 + 1, s28 >= 0, s28 <= s27 + 1, z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: O(n^2) [3*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [1 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 4*z'' + 2*z''^2], size: O(1) [0]
  encode_g: runtime: ?, size: O(n^1) [1 + z]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 11*z + 8*z^2

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

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

   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1 + s11 + s12 + 3*x_125 + 2*x_125^2 + 3*x_212 + 2*x_212^2 + 3*x_312 + 2*x_312^2 }-> s13 :|: s9 >= 0, s9 <= x_125, s10 >= 0, s10 <= x_212, s11 >= 0, s11 <= x_312, s12 >= 0, s12 <= 0, s13 >= 0, s13 <= s12 + 1, x_212 >= 0, x_312 >= 0, x_125 >= 0, z = 1 + (1 + x_125 + x_212 + x_312)
   encArg(z) -{ 2 + s14 + s15 + -5*z + 2*z^2 }-> s16 :|: s14 >= 0, s14 <= z - 2, s15 >= 0, s15 <= s14 + 1, s16 >= 0, s16 <= s15 + 1, z - 2 >= 0
   encArg(z) -{ 1 + s7 + 3*x_1 + 2*x_1^2 + 3*x_2 + 2*x_2^2 + 3*x_3 + 2*x_3^2 }-> s8 :|: s5 >= 0, s5 <= x_1, s6 >= 0, s6 <= x_2, s7 >= 0, s7 <= x_3, s8 >= 0, s8 <= 0, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z', z'') -{ 1 + s19 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 3*z'' + 2*z''^2 }-> s20 :|: s17 >= 0, s17 <= z, s18 >= 0, s18 <= z', s19 >= 0, s19 <= z'', s20 >= 0, s20 <= 0, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_g(z) -{ 1 + s23 + s24 + 3*x_153 + 2*x_153^2 + 3*x_226 + 2*x_226^2 + 3*x_326 + 2*x_326^2 }-> s25 :|: s21 >= 0, s21 <= x_153, s22 >= 0, s22 <= x_226, s23 >= 0, s23 <= x_326, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= s24 + 1, x_226 >= 0, z = 1 + x_153 + x_226 + x_326, x_153 >= 0, x_326 >= 0
   encode_g(z) -{ -1 + s26 + s27 + -1*z + 2*z^2 }-> s28 :|: s26 >= 0, s26 <= z - 1, s27 >= 0, s27 <= s26 + 1, s28 >= 0, s28 <= s27 + 1, z - 1 >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   f(z, z', z'') -{ 2 + s + z'' }-> s2 :|: s2 >= 0, s2 <= 0, s >= 0, s <= z'' - 1 + 1, z' >= 0, z'' - 1 >= 0, z = 1 + z'
   f(z, z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 2 }-> s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z'' >= 0, z = 1 + z'
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   g(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  g: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  f: runtime: O(n^1) [1 + z''], size: O(1) [0]
  encArg: runtime: O(n^2) [3*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [1 + 3*z + 2*z^2 + 3*z' + 2*z'^2 + 4*z'' + 2*z''^2], size: O(1) [0]
  encode_g: runtime: O(n^2) [1 + 11*z + 8*z^2], size: O(n^1) [1 + z]

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

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

(52)
BOUNDS(1, n^2)