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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) 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), 98 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxWeightedTrs
(11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedTrs
(13) CompletionProof [UPPER BOUND(ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTypedWeightedCompleteTrs
(17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms]
(18) CpxRNTS
(19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRNTS
(21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 322 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 1665 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 1189 ms]
(34) CpxRNTS
(35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 955 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 456 ms]
(40) CpxRNTS
(41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 179 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(46) CpxRNTS
(47) FinalProof [FINISHED, 0 ms]
(48) BOUNDS(1, n^2)


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

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


The TRS R consists of the following rules:

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

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

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


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

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

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

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

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

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

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

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

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

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

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

(6)
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(g(x, y), c_f(y, y)) -> f(g(y, x), y)

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

   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   f(x0, x1) -> c_f(x0, x1)

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

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

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

(8)
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, y), c_f(y, y)) -> f(g(y, x), y)

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

   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   f(x0, x1) -> c_f(x0, x1)

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

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

(10)
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(g(x, y), c_f(y, y)) -> f(g(y, x), y) [1]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(g(x, y), c_f(y, y)) -> f(g(y, x), y) [1]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]

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

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

(13) 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:

   encArg_1
   encode_f_2
   encode_g_2
   f_2

Due to the following rules being added:

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

And the following fresh constants:    const

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

(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(g(x, y), c_f(y, y)) -> f(g(y, x), y) [1]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   encode_g(v0, v1) -> const [0]
   f(v0, v1) -> const [0]

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

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

(16)
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(g(x, y), c_f(y, y)) -> f(g(y, x), y) [1]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(g(x_1', x_2'), g(x_11, x_21))) -> f(g(encArg(x_1'), encArg(x_2')), g(encArg(x_11), encArg(x_21))) [0]
   encArg(cons_f(g(x_1', x_2'), cons_f(x_12, x_22))) -> f(g(encArg(x_1'), encArg(x_2')), f(encArg(x_12), encArg(x_22))) [0]
   encArg(cons_f(g(x_1', x_2'), x_2)) -> f(g(encArg(x_1'), encArg(x_2')), const) [0]
   encArg(cons_f(cons_f(x_1'', x_2''), g(x_13, x_23))) -> f(f(encArg(x_1''), encArg(x_2'')), g(encArg(x_13), encArg(x_23))) [0]
   encArg(cons_f(cons_f(x_1'', x_2''), cons_f(x_14, x_24))) -> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) [0]
   encArg(cons_f(cons_f(x_1'', x_2''), x_2)) -> f(f(encArg(x_1''), encArg(x_2'')), const) [0]
   encArg(cons_f(x_1, g(x_15, x_25))) -> f(const, g(encArg(x_15), encArg(x_25))) [0]
   encArg(cons_f(x_1, cons_f(x_16, x_26))) -> f(const, f(encArg(x_16), encArg(x_26))) [0]
   encArg(cons_f(x_1, x_2)) -> f(const, const) [0]
   encode_f(g(x_17, x_27), g(x_19, x_29)) -> f(g(encArg(x_17), encArg(x_27)), g(encArg(x_19), encArg(x_29))) [0]
   encode_f(g(x_17, x_27), cons_f(x_110, x_210)) -> f(g(encArg(x_17), encArg(x_27)), f(encArg(x_110), encArg(x_210))) [0]
   encode_f(g(x_17, x_27), x_2) -> f(g(encArg(x_17), encArg(x_27)), const) [0]
   encode_f(cons_f(x_18, x_28), g(x_111, x_211)) -> f(f(encArg(x_18), encArg(x_28)), g(encArg(x_111), encArg(x_211))) [0]
   encode_f(cons_f(x_18, x_28), cons_f(x_112, x_212)) -> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) [0]
   encode_f(cons_f(x_18, x_28), x_2) -> f(f(encArg(x_18), encArg(x_28)), const) [0]
   encode_f(x_1, g(x_113, x_213)) -> f(const, g(encArg(x_113), encArg(x_213))) [0]
   encode_f(x_1, cons_f(x_114, x_214)) -> f(const, f(encArg(x_114), encArg(x_214))) [0]
   encode_f(x_1, x_2) -> f(const, const) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   encode_g(v0, v1) -> const [0]
   f(v0, v1) -> const [0]

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

Rewrite Strategy: INNERMOST
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(17) 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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, z' = x_2, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z = x_1, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: x_1 >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0, z = x_1
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_2 >= 0, z' = x_2
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   f(z, z') -{ 1 }-> f(1 + y + x, y) :|: z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1


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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(1 + y + x, y) :|: z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0


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

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

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(1 + y + x, y) :|: z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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

(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:

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(1 + y + x, y) :|: z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(1 + y + x, y) :|: z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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

(27) 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: z'

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> f(1 + y + x, y) :|: z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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

(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:

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encArg
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 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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

(33) 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 + 4*z^2

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

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

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), f(encArg(x_14), encArg(x_24))) :|: x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ 0 }-> f(0, f(encArg(x_16), encArg(x_26))) :|: x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), f(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1') + encArg(x_2'), 1 + encArg(x_11) + encArg(x_21)) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), f(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 0) :|: z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_114), encArg(x_214))) :|: z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_113) + encArg(x_213)) :|: z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), f(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 1 + encArg(z) + encArg(z') :|: z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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(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:

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ s11 + s12 + 3*x_1' + 4*x_1'^2 + 3*x_12 + 4*x_12^2 + 3*x_2' + 4*x_2'^2 + 3*x_22 + 4*x_22^2 }-> s13 :|: s8 >= 0, s8 <= x_1' + 1, s9 >= 0, s9 <= x_2' + 1, s10 >= 0, s10 <= x_12 + 1, s11 >= 0, s11 <= x_22 + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= 1 + s8 + s9 + s12 + 1, x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 3*x_1' + 4*x_1'^2 + 3*x_2' + 4*x_2'^2 }-> s16 :|: s14 >= 0, s14 <= x_1' + 1, s15 >= 0, s15 <= x_2' + 1, s16 >= 0, s16 <= 1 + s14 + s15 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s18 + s20 + s21 + 3*x_1'' + 4*x_1''^2 + 3*x_13 + 4*x_13^2 + 3*x_2'' + 4*x_2''^2 + 3*x_23 + 4*x_23^2 }-> s22 :|: s17 >= 0, s17 <= x_1'' + 1, s18 >= 0, s18 <= x_2'' + 1, s19 >= 0, s19 <= s17 + s18 + 1, s20 >= 0, s20 <= x_13 + 1, s21 >= 0, s21 <= x_23 + 1, s22 >= 0, s22 <= s19 + (1 + s20 + s21) + 1, x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ s24 + s27 + s28 + 3*x_1'' + 4*x_1''^2 + 3*x_14 + 4*x_14^2 + 3*x_2'' + 4*x_2''^2 + 3*x_24 + 4*x_24^2 }-> s29 :|: s23 >= 0, s23 <= x_1'' + 1, s24 >= 0, s24 <= x_2'' + 1, s25 >= 0, s25 <= s23 + s24 + 1, s26 >= 0, s26 <= x_14 + 1, s27 >= 0, s27 <= x_24 + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s25 + s28 + 1, x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ s31 + 3*x_1'' + 4*x_1''^2 + 3*x_2'' + 4*x_2''^2 }-> s33 :|: s30 >= 0, s30 <= x_1'' + 1, s31 >= 0, s31 <= x_2'' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= s32 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s34 + s35 + 3*x_15 + 4*x_15^2 + 3*x_25 + 4*x_25^2 }-> s36 :|: s34 >= 0, s34 <= x_15 + 1, s35 >= 0, s35 <= x_25 + 1, s36 >= 0, s36 <= 0 + (1 + s34 + s35) + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ s38 + s39 + 3*x_16 + 4*x_16^2 + 3*x_26 + 4*x_26^2 }-> s40 :|: s37 >= 0, s37 <= x_16 + 1, s38 >= 0, s38 <= x_26 + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= 0 + s39 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 1 + s5 + s6 + 3*x_1' + 4*x_1'^2 + 3*x_11 + 4*x_11^2 + 3*x_2' + 4*x_2'^2 + 3*x_21 + 4*x_21^2 }-> s7 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= x_11 + 1, s6 >= 0, s6 <= x_21 + 1, s7 >= 0, s7 <= 1 + s3 + s4 + (1 + s5 + s6) + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 3*x_1 + 4*x_1^2 + 3*x_2 + 4*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s43 + s44 + 3*x_17 + 4*x_17^2 + 3*x_19 + 4*x_19^2 + 3*x_27 + 4*x_27^2 + 3*x_29 + 4*x_29^2 }-> s45 :|: s41 >= 0, s41 <= x_17 + 1, s42 >= 0, s42 <= x_27 + 1, s43 >= 0, s43 <= x_19 + 1, s44 >= 0, s44 <= x_29 + 1, s45 >= 0, s45 <= 1 + s41 + s42 + (1 + s43 + s44) + 1, z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ s49 + s50 + 3*x_110 + 4*x_110^2 + 3*x_17 + 4*x_17^2 + 3*x_210 + 4*x_210^2 + 3*x_27 + 4*x_27^2 }-> s51 :|: s46 >= 0, s46 <= x_17 + 1, s47 >= 0, s47 <= x_27 + 1, s48 >= 0, s48 <= x_110 + 1, s49 >= 0, s49 <= x_210 + 1, s50 >= 0, s50 <= s48 + s49 + 1, s51 >= 0, s51 <= 1 + s46 + s47 + s50 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 3*x_17 + 4*x_17^2 + 3*x_27 + 4*x_27^2 }-> s54 :|: s52 >= 0, s52 <= x_17 + 1, s53 >= 0, s53 <= x_27 + 1, s54 >= 0, s54 <= 1 + s52 + s53 + 0 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s56 + s58 + s59 + 3*x_111 + 4*x_111^2 + 3*x_18 + 4*x_18^2 + 3*x_211 + 4*x_211^2 + 3*x_28 + 4*x_28^2 }-> s60 :|: s55 >= 0, s55 <= x_18 + 1, s56 >= 0, s56 <= x_28 + 1, s57 >= 0, s57 <= s55 + s56 + 1, s58 >= 0, s58 <= x_111 + 1, s59 >= 0, s59 <= x_211 + 1, s60 >= 0, s60 <= s57 + (1 + s58 + s59) + 1, z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ s62 + s65 + s66 + 3*x_112 + 4*x_112^2 + 3*x_18 + 4*x_18^2 + 3*x_212 + 4*x_212^2 + 3*x_28 + 4*x_28^2 }-> s67 :|: s61 >= 0, s61 <= x_18 + 1, s62 >= 0, s62 <= x_28 + 1, s63 >= 0, s63 <= s61 + s62 + 1, s64 >= 0, s64 <= x_112 + 1, s65 >= 0, s65 <= x_212 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= s63 + s66 + 1, x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ s69 + 3*x_18 + 4*x_18^2 + 3*x_28 + 4*x_28^2 }-> s71 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= x_28 + 1, s70 >= 0, s70 <= s68 + s69 + 1, s71 >= 0, s71 <= s70 + 0 + 1, z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 1 + s72 + s73 + 3*x_113 + 4*x_113^2 + 3*x_213 + 4*x_213^2 }-> s74 :|: s72 >= 0, s72 <= x_113 + 1, s73 >= 0, s73 <= x_213 + 1, s74 >= 0, s74 <= 0 + (1 + s72 + s73) + 1, z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ s76 + s77 + 3*x_114 + 4*x_114^2 + 3*x_214 + 4*x_214^2 }-> s78 :|: s75 >= 0, s75 <= x_114 + 1, s76 >= 0, s76 <= x_214 + 1, s77 >= 0, s77 <= s75 + s76 + 1, s78 >= 0, s78 <= 0 + s77 + 1, z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 3*z + 4*z^2 + 3*z' + 4*z'^2 }-> 1 + s79 + s80 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + z + z'

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

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

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ s11 + s12 + 3*x_1' + 4*x_1'^2 + 3*x_12 + 4*x_12^2 + 3*x_2' + 4*x_2'^2 + 3*x_22 + 4*x_22^2 }-> s13 :|: s8 >= 0, s8 <= x_1' + 1, s9 >= 0, s9 <= x_2' + 1, s10 >= 0, s10 <= x_12 + 1, s11 >= 0, s11 <= x_22 + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= 1 + s8 + s9 + s12 + 1, x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 3*x_1' + 4*x_1'^2 + 3*x_2' + 4*x_2'^2 }-> s16 :|: s14 >= 0, s14 <= x_1' + 1, s15 >= 0, s15 <= x_2' + 1, s16 >= 0, s16 <= 1 + s14 + s15 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s18 + s20 + s21 + 3*x_1'' + 4*x_1''^2 + 3*x_13 + 4*x_13^2 + 3*x_2'' + 4*x_2''^2 + 3*x_23 + 4*x_23^2 }-> s22 :|: s17 >= 0, s17 <= x_1'' + 1, s18 >= 0, s18 <= x_2'' + 1, s19 >= 0, s19 <= s17 + s18 + 1, s20 >= 0, s20 <= x_13 + 1, s21 >= 0, s21 <= x_23 + 1, s22 >= 0, s22 <= s19 + (1 + s20 + s21) + 1, x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ s24 + s27 + s28 + 3*x_1'' + 4*x_1''^2 + 3*x_14 + 4*x_14^2 + 3*x_2'' + 4*x_2''^2 + 3*x_24 + 4*x_24^2 }-> s29 :|: s23 >= 0, s23 <= x_1'' + 1, s24 >= 0, s24 <= x_2'' + 1, s25 >= 0, s25 <= s23 + s24 + 1, s26 >= 0, s26 <= x_14 + 1, s27 >= 0, s27 <= x_24 + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s25 + s28 + 1, x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ s31 + 3*x_1'' + 4*x_1''^2 + 3*x_2'' + 4*x_2''^2 }-> s33 :|: s30 >= 0, s30 <= x_1'' + 1, s31 >= 0, s31 <= x_2'' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= s32 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s34 + s35 + 3*x_15 + 4*x_15^2 + 3*x_25 + 4*x_25^2 }-> s36 :|: s34 >= 0, s34 <= x_15 + 1, s35 >= 0, s35 <= x_25 + 1, s36 >= 0, s36 <= 0 + (1 + s34 + s35) + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ s38 + s39 + 3*x_16 + 4*x_16^2 + 3*x_26 + 4*x_26^2 }-> s40 :|: s37 >= 0, s37 <= x_16 + 1, s38 >= 0, s38 <= x_26 + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= 0 + s39 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 1 + s5 + s6 + 3*x_1' + 4*x_1'^2 + 3*x_11 + 4*x_11^2 + 3*x_2' + 4*x_2'^2 + 3*x_21 + 4*x_21^2 }-> s7 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= x_11 + 1, s6 >= 0, s6 <= x_21 + 1, s7 >= 0, s7 <= 1 + s3 + s4 + (1 + s5 + s6) + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 3*x_1 + 4*x_1^2 + 3*x_2 + 4*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s43 + s44 + 3*x_17 + 4*x_17^2 + 3*x_19 + 4*x_19^2 + 3*x_27 + 4*x_27^2 + 3*x_29 + 4*x_29^2 }-> s45 :|: s41 >= 0, s41 <= x_17 + 1, s42 >= 0, s42 <= x_27 + 1, s43 >= 0, s43 <= x_19 + 1, s44 >= 0, s44 <= x_29 + 1, s45 >= 0, s45 <= 1 + s41 + s42 + (1 + s43 + s44) + 1, z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ s49 + s50 + 3*x_110 + 4*x_110^2 + 3*x_17 + 4*x_17^2 + 3*x_210 + 4*x_210^2 + 3*x_27 + 4*x_27^2 }-> s51 :|: s46 >= 0, s46 <= x_17 + 1, s47 >= 0, s47 <= x_27 + 1, s48 >= 0, s48 <= x_110 + 1, s49 >= 0, s49 <= x_210 + 1, s50 >= 0, s50 <= s48 + s49 + 1, s51 >= 0, s51 <= 1 + s46 + s47 + s50 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 3*x_17 + 4*x_17^2 + 3*x_27 + 4*x_27^2 }-> s54 :|: s52 >= 0, s52 <= x_17 + 1, s53 >= 0, s53 <= x_27 + 1, s54 >= 0, s54 <= 1 + s52 + s53 + 0 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s56 + s58 + s59 + 3*x_111 + 4*x_111^2 + 3*x_18 + 4*x_18^2 + 3*x_211 + 4*x_211^2 + 3*x_28 + 4*x_28^2 }-> s60 :|: s55 >= 0, s55 <= x_18 + 1, s56 >= 0, s56 <= x_28 + 1, s57 >= 0, s57 <= s55 + s56 + 1, s58 >= 0, s58 <= x_111 + 1, s59 >= 0, s59 <= x_211 + 1, s60 >= 0, s60 <= s57 + (1 + s58 + s59) + 1, z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ s62 + s65 + s66 + 3*x_112 + 4*x_112^2 + 3*x_18 + 4*x_18^2 + 3*x_212 + 4*x_212^2 + 3*x_28 + 4*x_28^2 }-> s67 :|: s61 >= 0, s61 <= x_18 + 1, s62 >= 0, s62 <= x_28 + 1, s63 >= 0, s63 <= s61 + s62 + 1, s64 >= 0, s64 <= x_112 + 1, s65 >= 0, s65 <= x_212 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= s63 + s66 + 1, x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ s69 + 3*x_18 + 4*x_18^2 + 3*x_28 + 4*x_28^2 }-> s71 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= x_28 + 1, s70 >= 0, s70 <= s68 + s69 + 1, s71 >= 0, s71 <= s70 + 0 + 1, z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 1 + s72 + s73 + 3*x_113 + 4*x_113^2 + 3*x_213 + 4*x_213^2 }-> s74 :|: s72 >= 0, s72 <= x_113 + 1, s73 >= 0, s73 <= x_213 + 1, s74 >= 0, s74 <= 0 + (1 + s72 + s73) + 1, z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ s76 + s77 + 3*x_114 + 4*x_114^2 + 3*x_214 + 4*x_214^2 }-> s78 :|: s75 >= 0, s75 <= x_114 + 1, s76 >= 0, s76 <= x_214 + 1, s77 >= 0, s77 <= s75 + s76 + 1, s78 >= 0, s78 <= 0 + s77 + 1, z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 3*z + 4*z^2 + 3*z' + 4*z'^2 }-> 1 + s79 + s80 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [z'], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^2) [3*z + 4*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: ?, size: O(n^1) [5 + z + z']

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

(39) 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: 18 + 39*z + 48*z^2 + 51*z' + 48*z'^2

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

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

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ s11 + s12 + 3*x_1' + 4*x_1'^2 + 3*x_12 + 4*x_12^2 + 3*x_2' + 4*x_2'^2 + 3*x_22 + 4*x_22^2 }-> s13 :|: s8 >= 0, s8 <= x_1' + 1, s9 >= 0, s9 <= x_2' + 1, s10 >= 0, s10 <= x_12 + 1, s11 >= 0, s11 <= x_22 + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= 1 + s8 + s9 + s12 + 1, x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 3*x_1' + 4*x_1'^2 + 3*x_2' + 4*x_2'^2 }-> s16 :|: s14 >= 0, s14 <= x_1' + 1, s15 >= 0, s15 <= x_2' + 1, s16 >= 0, s16 <= 1 + s14 + s15 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s18 + s20 + s21 + 3*x_1'' + 4*x_1''^2 + 3*x_13 + 4*x_13^2 + 3*x_2'' + 4*x_2''^2 + 3*x_23 + 4*x_23^2 }-> s22 :|: s17 >= 0, s17 <= x_1'' + 1, s18 >= 0, s18 <= x_2'' + 1, s19 >= 0, s19 <= s17 + s18 + 1, s20 >= 0, s20 <= x_13 + 1, s21 >= 0, s21 <= x_23 + 1, s22 >= 0, s22 <= s19 + (1 + s20 + s21) + 1, x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ s24 + s27 + s28 + 3*x_1'' + 4*x_1''^2 + 3*x_14 + 4*x_14^2 + 3*x_2'' + 4*x_2''^2 + 3*x_24 + 4*x_24^2 }-> s29 :|: s23 >= 0, s23 <= x_1'' + 1, s24 >= 0, s24 <= x_2'' + 1, s25 >= 0, s25 <= s23 + s24 + 1, s26 >= 0, s26 <= x_14 + 1, s27 >= 0, s27 <= x_24 + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s25 + s28 + 1, x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ s31 + 3*x_1'' + 4*x_1''^2 + 3*x_2'' + 4*x_2''^2 }-> s33 :|: s30 >= 0, s30 <= x_1'' + 1, s31 >= 0, s31 <= x_2'' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= s32 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s34 + s35 + 3*x_15 + 4*x_15^2 + 3*x_25 + 4*x_25^2 }-> s36 :|: s34 >= 0, s34 <= x_15 + 1, s35 >= 0, s35 <= x_25 + 1, s36 >= 0, s36 <= 0 + (1 + s34 + s35) + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ s38 + s39 + 3*x_16 + 4*x_16^2 + 3*x_26 + 4*x_26^2 }-> s40 :|: s37 >= 0, s37 <= x_16 + 1, s38 >= 0, s38 <= x_26 + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= 0 + s39 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 1 + s5 + s6 + 3*x_1' + 4*x_1'^2 + 3*x_11 + 4*x_11^2 + 3*x_2' + 4*x_2'^2 + 3*x_21 + 4*x_21^2 }-> s7 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= x_11 + 1, s6 >= 0, s6 <= x_21 + 1, s7 >= 0, s7 <= 1 + s3 + s4 + (1 + s5 + s6) + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 3*x_1 + 4*x_1^2 + 3*x_2 + 4*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s43 + s44 + 3*x_17 + 4*x_17^2 + 3*x_19 + 4*x_19^2 + 3*x_27 + 4*x_27^2 + 3*x_29 + 4*x_29^2 }-> s45 :|: s41 >= 0, s41 <= x_17 + 1, s42 >= 0, s42 <= x_27 + 1, s43 >= 0, s43 <= x_19 + 1, s44 >= 0, s44 <= x_29 + 1, s45 >= 0, s45 <= 1 + s41 + s42 + (1 + s43 + s44) + 1, z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ s49 + s50 + 3*x_110 + 4*x_110^2 + 3*x_17 + 4*x_17^2 + 3*x_210 + 4*x_210^2 + 3*x_27 + 4*x_27^2 }-> s51 :|: s46 >= 0, s46 <= x_17 + 1, s47 >= 0, s47 <= x_27 + 1, s48 >= 0, s48 <= x_110 + 1, s49 >= 0, s49 <= x_210 + 1, s50 >= 0, s50 <= s48 + s49 + 1, s51 >= 0, s51 <= 1 + s46 + s47 + s50 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 3*x_17 + 4*x_17^2 + 3*x_27 + 4*x_27^2 }-> s54 :|: s52 >= 0, s52 <= x_17 + 1, s53 >= 0, s53 <= x_27 + 1, s54 >= 0, s54 <= 1 + s52 + s53 + 0 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s56 + s58 + s59 + 3*x_111 + 4*x_111^2 + 3*x_18 + 4*x_18^2 + 3*x_211 + 4*x_211^2 + 3*x_28 + 4*x_28^2 }-> s60 :|: s55 >= 0, s55 <= x_18 + 1, s56 >= 0, s56 <= x_28 + 1, s57 >= 0, s57 <= s55 + s56 + 1, s58 >= 0, s58 <= x_111 + 1, s59 >= 0, s59 <= x_211 + 1, s60 >= 0, s60 <= s57 + (1 + s58 + s59) + 1, z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ s62 + s65 + s66 + 3*x_112 + 4*x_112^2 + 3*x_18 + 4*x_18^2 + 3*x_212 + 4*x_212^2 + 3*x_28 + 4*x_28^2 }-> s67 :|: s61 >= 0, s61 <= x_18 + 1, s62 >= 0, s62 <= x_28 + 1, s63 >= 0, s63 <= s61 + s62 + 1, s64 >= 0, s64 <= x_112 + 1, s65 >= 0, s65 <= x_212 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= s63 + s66 + 1, x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ s69 + 3*x_18 + 4*x_18^2 + 3*x_28 + 4*x_28^2 }-> s71 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= x_28 + 1, s70 >= 0, s70 <= s68 + s69 + 1, s71 >= 0, s71 <= s70 + 0 + 1, z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 1 + s72 + s73 + 3*x_113 + 4*x_113^2 + 3*x_213 + 4*x_213^2 }-> s74 :|: s72 >= 0, s72 <= x_113 + 1, s73 >= 0, s73 <= x_213 + 1, s74 >= 0, s74 <= 0 + (1 + s72 + s73) + 1, z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ s76 + s77 + 3*x_114 + 4*x_114^2 + 3*x_214 + 4*x_214^2 }-> s78 :|: s75 >= 0, s75 <= x_114 + 1, s76 >= 0, s76 <= x_214 + 1, s77 >= 0, s77 <= s75 + s76 + 1, s78 >= 0, s78 <= 0 + s77 + 1, z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 3*z + 4*z^2 + 3*z' + 4*z'^2 }-> 1 + s79 + s80 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [z'], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^2) [3*z + 4*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [18 + 39*z + 48*z^2 + 51*z' + 48*z'^2], size: O(n^1) [5 + z + z']

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(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:

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ s11 + s12 + 3*x_1' + 4*x_1'^2 + 3*x_12 + 4*x_12^2 + 3*x_2' + 4*x_2'^2 + 3*x_22 + 4*x_22^2 }-> s13 :|: s8 >= 0, s8 <= x_1' + 1, s9 >= 0, s9 <= x_2' + 1, s10 >= 0, s10 <= x_12 + 1, s11 >= 0, s11 <= x_22 + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= 1 + s8 + s9 + s12 + 1, x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 3*x_1' + 4*x_1'^2 + 3*x_2' + 4*x_2'^2 }-> s16 :|: s14 >= 0, s14 <= x_1' + 1, s15 >= 0, s15 <= x_2' + 1, s16 >= 0, s16 <= 1 + s14 + s15 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s18 + s20 + s21 + 3*x_1'' + 4*x_1''^2 + 3*x_13 + 4*x_13^2 + 3*x_2'' + 4*x_2''^2 + 3*x_23 + 4*x_23^2 }-> s22 :|: s17 >= 0, s17 <= x_1'' + 1, s18 >= 0, s18 <= x_2'' + 1, s19 >= 0, s19 <= s17 + s18 + 1, s20 >= 0, s20 <= x_13 + 1, s21 >= 0, s21 <= x_23 + 1, s22 >= 0, s22 <= s19 + (1 + s20 + s21) + 1, x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ s24 + s27 + s28 + 3*x_1'' + 4*x_1''^2 + 3*x_14 + 4*x_14^2 + 3*x_2'' + 4*x_2''^2 + 3*x_24 + 4*x_24^2 }-> s29 :|: s23 >= 0, s23 <= x_1'' + 1, s24 >= 0, s24 <= x_2'' + 1, s25 >= 0, s25 <= s23 + s24 + 1, s26 >= 0, s26 <= x_14 + 1, s27 >= 0, s27 <= x_24 + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s25 + s28 + 1, x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ s31 + 3*x_1'' + 4*x_1''^2 + 3*x_2'' + 4*x_2''^2 }-> s33 :|: s30 >= 0, s30 <= x_1'' + 1, s31 >= 0, s31 <= x_2'' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= s32 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s34 + s35 + 3*x_15 + 4*x_15^2 + 3*x_25 + 4*x_25^2 }-> s36 :|: s34 >= 0, s34 <= x_15 + 1, s35 >= 0, s35 <= x_25 + 1, s36 >= 0, s36 <= 0 + (1 + s34 + s35) + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ s38 + s39 + 3*x_16 + 4*x_16^2 + 3*x_26 + 4*x_26^2 }-> s40 :|: s37 >= 0, s37 <= x_16 + 1, s38 >= 0, s38 <= x_26 + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= 0 + s39 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 1 + s5 + s6 + 3*x_1' + 4*x_1'^2 + 3*x_11 + 4*x_11^2 + 3*x_2' + 4*x_2'^2 + 3*x_21 + 4*x_21^2 }-> s7 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= x_11 + 1, s6 >= 0, s6 <= x_21 + 1, s7 >= 0, s7 <= 1 + s3 + s4 + (1 + s5 + s6) + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 3*x_1 + 4*x_1^2 + 3*x_2 + 4*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s43 + s44 + 3*x_17 + 4*x_17^2 + 3*x_19 + 4*x_19^2 + 3*x_27 + 4*x_27^2 + 3*x_29 + 4*x_29^2 }-> s45 :|: s41 >= 0, s41 <= x_17 + 1, s42 >= 0, s42 <= x_27 + 1, s43 >= 0, s43 <= x_19 + 1, s44 >= 0, s44 <= x_29 + 1, s45 >= 0, s45 <= 1 + s41 + s42 + (1 + s43 + s44) + 1, z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ s49 + s50 + 3*x_110 + 4*x_110^2 + 3*x_17 + 4*x_17^2 + 3*x_210 + 4*x_210^2 + 3*x_27 + 4*x_27^2 }-> s51 :|: s46 >= 0, s46 <= x_17 + 1, s47 >= 0, s47 <= x_27 + 1, s48 >= 0, s48 <= x_110 + 1, s49 >= 0, s49 <= x_210 + 1, s50 >= 0, s50 <= s48 + s49 + 1, s51 >= 0, s51 <= 1 + s46 + s47 + s50 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 3*x_17 + 4*x_17^2 + 3*x_27 + 4*x_27^2 }-> s54 :|: s52 >= 0, s52 <= x_17 + 1, s53 >= 0, s53 <= x_27 + 1, s54 >= 0, s54 <= 1 + s52 + s53 + 0 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s56 + s58 + s59 + 3*x_111 + 4*x_111^2 + 3*x_18 + 4*x_18^2 + 3*x_211 + 4*x_211^2 + 3*x_28 + 4*x_28^2 }-> s60 :|: s55 >= 0, s55 <= x_18 + 1, s56 >= 0, s56 <= x_28 + 1, s57 >= 0, s57 <= s55 + s56 + 1, s58 >= 0, s58 <= x_111 + 1, s59 >= 0, s59 <= x_211 + 1, s60 >= 0, s60 <= s57 + (1 + s58 + s59) + 1, z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ s62 + s65 + s66 + 3*x_112 + 4*x_112^2 + 3*x_18 + 4*x_18^2 + 3*x_212 + 4*x_212^2 + 3*x_28 + 4*x_28^2 }-> s67 :|: s61 >= 0, s61 <= x_18 + 1, s62 >= 0, s62 <= x_28 + 1, s63 >= 0, s63 <= s61 + s62 + 1, s64 >= 0, s64 <= x_112 + 1, s65 >= 0, s65 <= x_212 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= s63 + s66 + 1, x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ s69 + 3*x_18 + 4*x_18^2 + 3*x_28 + 4*x_28^2 }-> s71 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= x_28 + 1, s70 >= 0, s70 <= s68 + s69 + 1, s71 >= 0, s71 <= s70 + 0 + 1, z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 1 + s72 + s73 + 3*x_113 + 4*x_113^2 + 3*x_213 + 4*x_213^2 }-> s74 :|: s72 >= 0, s72 <= x_113 + 1, s73 >= 0, s73 <= x_213 + 1, s74 >= 0, s74 <= 0 + (1 + s72 + s73) + 1, z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ s76 + s77 + 3*x_114 + 4*x_114^2 + 3*x_214 + 4*x_214^2 }-> s78 :|: s75 >= 0, s75 <= x_114 + 1, s76 >= 0, s76 <= x_214 + 1, s77 >= 0, s77 <= s75 + s76 + 1, s78 >= 0, s78 <= 0 + s77 + 1, z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 3*z + 4*z^2 + 3*z' + 4*z'^2 }-> 1 + s79 + s80 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [z'], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^2) [3*z + 4*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [18 + 39*z + 48*z^2 + 51*z' + 48*z'^2], size: O(n^1) [5 + z + z']

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(43) 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: 3 + z + z'

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

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

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ s11 + s12 + 3*x_1' + 4*x_1'^2 + 3*x_12 + 4*x_12^2 + 3*x_2' + 4*x_2'^2 + 3*x_22 + 4*x_22^2 }-> s13 :|: s8 >= 0, s8 <= x_1' + 1, s9 >= 0, s9 <= x_2' + 1, s10 >= 0, s10 <= x_12 + 1, s11 >= 0, s11 <= x_22 + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= 1 + s8 + s9 + s12 + 1, x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 3*x_1' + 4*x_1'^2 + 3*x_2' + 4*x_2'^2 }-> s16 :|: s14 >= 0, s14 <= x_1' + 1, s15 >= 0, s15 <= x_2' + 1, s16 >= 0, s16 <= 1 + s14 + s15 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s18 + s20 + s21 + 3*x_1'' + 4*x_1''^2 + 3*x_13 + 4*x_13^2 + 3*x_2'' + 4*x_2''^2 + 3*x_23 + 4*x_23^2 }-> s22 :|: s17 >= 0, s17 <= x_1'' + 1, s18 >= 0, s18 <= x_2'' + 1, s19 >= 0, s19 <= s17 + s18 + 1, s20 >= 0, s20 <= x_13 + 1, s21 >= 0, s21 <= x_23 + 1, s22 >= 0, s22 <= s19 + (1 + s20 + s21) + 1, x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ s24 + s27 + s28 + 3*x_1'' + 4*x_1''^2 + 3*x_14 + 4*x_14^2 + 3*x_2'' + 4*x_2''^2 + 3*x_24 + 4*x_24^2 }-> s29 :|: s23 >= 0, s23 <= x_1'' + 1, s24 >= 0, s24 <= x_2'' + 1, s25 >= 0, s25 <= s23 + s24 + 1, s26 >= 0, s26 <= x_14 + 1, s27 >= 0, s27 <= x_24 + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s25 + s28 + 1, x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ s31 + 3*x_1'' + 4*x_1''^2 + 3*x_2'' + 4*x_2''^2 }-> s33 :|: s30 >= 0, s30 <= x_1'' + 1, s31 >= 0, s31 <= x_2'' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= s32 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s34 + s35 + 3*x_15 + 4*x_15^2 + 3*x_25 + 4*x_25^2 }-> s36 :|: s34 >= 0, s34 <= x_15 + 1, s35 >= 0, s35 <= x_25 + 1, s36 >= 0, s36 <= 0 + (1 + s34 + s35) + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ s38 + s39 + 3*x_16 + 4*x_16^2 + 3*x_26 + 4*x_26^2 }-> s40 :|: s37 >= 0, s37 <= x_16 + 1, s38 >= 0, s38 <= x_26 + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= 0 + s39 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 1 + s5 + s6 + 3*x_1' + 4*x_1'^2 + 3*x_11 + 4*x_11^2 + 3*x_2' + 4*x_2'^2 + 3*x_21 + 4*x_21^2 }-> s7 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= x_11 + 1, s6 >= 0, s6 <= x_21 + 1, s7 >= 0, s7 <= 1 + s3 + s4 + (1 + s5 + s6) + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 3*x_1 + 4*x_1^2 + 3*x_2 + 4*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s43 + s44 + 3*x_17 + 4*x_17^2 + 3*x_19 + 4*x_19^2 + 3*x_27 + 4*x_27^2 + 3*x_29 + 4*x_29^2 }-> s45 :|: s41 >= 0, s41 <= x_17 + 1, s42 >= 0, s42 <= x_27 + 1, s43 >= 0, s43 <= x_19 + 1, s44 >= 0, s44 <= x_29 + 1, s45 >= 0, s45 <= 1 + s41 + s42 + (1 + s43 + s44) + 1, z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ s49 + s50 + 3*x_110 + 4*x_110^2 + 3*x_17 + 4*x_17^2 + 3*x_210 + 4*x_210^2 + 3*x_27 + 4*x_27^2 }-> s51 :|: s46 >= 0, s46 <= x_17 + 1, s47 >= 0, s47 <= x_27 + 1, s48 >= 0, s48 <= x_110 + 1, s49 >= 0, s49 <= x_210 + 1, s50 >= 0, s50 <= s48 + s49 + 1, s51 >= 0, s51 <= 1 + s46 + s47 + s50 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 3*x_17 + 4*x_17^2 + 3*x_27 + 4*x_27^2 }-> s54 :|: s52 >= 0, s52 <= x_17 + 1, s53 >= 0, s53 <= x_27 + 1, s54 >= 0, s54 <= 1 + s52 + s53 + 0 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s56 + s58 + s59 + 3*x_111 + 4*x_111^2 + 3*x_18 + 4*x_18^2 + 3*x_211 + 4*x_211^2 + 3*x_28 + 4*x_28^2 }-> s60 :|: s55 >= 0, s55 <= x_18 + 1, s56 >= 0, s56 <= x_28 + 1, s57 >= 0, s57 <= s55 + s56 + 1, s58 >= 0, s58 <= x_111 + 1, s59 >= 0, s59 <= x_211 + 1, s60 >= 0, s60 <= s57 + (1 + s58 + s59) + 1, z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ s62 + s65 + s66 + 3*x_112 + 4*x_112^2 + 3*x_18 + 4*x_18^2 + 3*x_212 + 4*x_212^2 + 3*x_28 + 4*x_28^2 }-> s67 :|: s61 >= 0, s61 <= x_18 + 1, s62 >= 0, s62 <= x_28 + 1, s63 >= 0, s63 <= s61 + s62 + 1, s64 >= 0, s64 <= x_112 + 1, s65 >= 0, s65 <= x_212 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= s63 + s66 + 1, x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ s69 + 3*x_18 + 4*x_18^2 + 3*x_28 + 4*x_28^2 }-> s71 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= x_28 + 1, s70 >= 0, s70 <= s68 + s69 + 1, s71 >= 0, s71 <= s70 + 0 + 1, z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 1 + s72 + s73 + 3*x_113 + 4*x_113^2 + 3*x_213 + 4*x_213^2 }-> s74 :|: s72 >= 0, s72 <= x_113 + 1, s73 >= 0, s73 <= x_213 + 1, s74 >= 0, s74 <= 0 + (1 + s72 + s73) + 1, z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ s76 + s77 + 3*x_114 + 4*x_114^2 + 3*x_214 + 4*x_214^2 }-> s78 :|: s75 >= 0, s75 <= x_114 + 1, s76 >= 0, s76 <= x_214 + 1, s77 >= 0, s77 <= s75 + s76 + 1, s78 >= 0, s78 <= 0 + s77 + 1, z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 3*z + 4*z^2 + 3*z' + 4*z'^2 }-> 1 + s79 + s80 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [z'], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^2) [3*z + 4*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [18 + 39*z + 48*z^2 + 51*z' + 48*z'^2], size: O(n^1) [5 + z + z']
  encode_g: runtime: ?, size: O(n^1) [3 + z + z']

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(45) 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: 3*z + 4*z^2 + 3*z' + 4*z'^2

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

   encArg(z) -{ 0 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ s11 + s12 + 3*x_1' + 4*x_1'^2 + 3*x_12 + 4*x_12^2 + 3*x_2' + 4*x_2'^2 + 3*x_22 + 4*x_22^2 }-> s13 :|: s8 >= 0, s8 <= x_1' + 1, s9 >= 0, s9 <= x_2' + 1, s10 >= 0, s10 <= x_12 + 1, s11 >= 0, s11 <= x_22 + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= 1 + s8 + s9 + s12 + 1, x_2' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_1' >= 0, x_12 >= 0, x_22 >= 0
   encArg(z) -{ 3*x_1' + 4*x_1'^2 + 3*x_2' + 4*x_2'^2 }-> s16 :|: s14 >= 0, s14 <= x_1' + 1, s15 >= 0, s15 <= x_2' + 1, s16 >= 0, s16 <= 1 + s14 + s15 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s18 + s20 + s21 + 3*x_1'' + 4*x_1''^2 + 3*x_13 + 4*x_13^2 + 3*x_2'' + 4*x_2''^2 + 3*x_23 + 4*x_23^2 }-> s22 :|: s17 >= 0, s17 <= x_1'' + 1, s18 >= 0, s18 <= x_2'' + 1, s19 >= 0, s19 <= s17 + s18 + 1, s20 >= 0, s20 <= x_13 + 1, s21 >= 0, s21 <= x_23 + 1, s22 >= 0, s22 <= s19 + (1 + s20 + s21) + 1, x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z) -{ s24 + s27 + s28 + 3*x_1'' + 4*x_1''^2 + 3*x_14 + 4*x_14^2 + 3*x_2'' + 4*x_2''^2 + 3*x_24 + 4*x_24^2 }-> s29 :|: s23 >= 0, s23 <= x_1'' + 1, s24 >= 0, s24 <= x_2'' + 1, s25 >= 0, s25 <= s23 + s24 + 1, s26 >= 0, s26 <= x_14 + 1, s27 >= 0, s27 <= x_24 + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s25 + s28 + 1, x_1'' >= 0, x_14 >= 0, x_2'' >= 0, x_24 >= 0, z = 1 + (1 + x_1'' + x_2'') + (1 + x_14 + x_24)
   encArg(z) -{ s31 + 3*x_1'' + 4*x_1''^2 + 3*x_2'' + 4*x_2''^2 }-> s33 :|: s30 >= 0, s30 <= x_1'' + 1, s31 >= 0, s31 <= x_2'' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= s32 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'' + x_2'') + x_2, x_2'' >= 0, x_2 >= 0
   encArg(z) -{ 1 + s34 + s35 + 3*x_15 + 4*x_15^2 + 3*x_25 + 4*x_25^2 }-> s36 :|: s34 >= 0, s34 <= x_15 + 1, s35 >= 0, s35 <= x_25 + 1, s36 >= 0, s36 <= 0 + (1 + s34 + s35) + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z) -{ s38 + s39 + 3*x_16 + 4*x_16^2 + 3*x_26 + 4*x_26^2 }-> s40 :|: s37 >= 0, s37 <= x_16 + 1, s38 >= 0, s38 <= x_26 + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= 0 + s39 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z) -{ 1 + s5 + s6 + 3*x_1' + 4*x_1'^2 + 3*x_11 + 4*x_11^2 + 3*x_2' + 4*x_2'^2 + 3*x_21 + 4*x_21^2 }-> s7 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= x_11 + 1, s6 >= 0, s6 <= x_21 + 1, s7 >= 0, s7 <= 1 + s3 + s4 + (1 + s5 + s6) + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, x_21 >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21)
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 3*x_1 + 4*x_1^2 + 3*x_2 + 4*x_2^2 }-> 1 + s1 + s2 :|: s1 >= 0, s1 <= x_1 + 1, s2 >= 0, s2 <= x_2 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s43 + s44 + 3*x_17 + 4*x_17^2 + 3*x_19 + 4*x_19^2 + 3*x_27 + 4*x_27^2 + 3*x_29 + 4*x_29^2 }-> s45 :|: s41 >= 0, s41 <= x_17 + 1, s42 >= 0, s42 <= x_27 + 1, s43 >= 0, s43 <= x_19 + 1, s44 >= 0, s44 <= x_29 + 1, s45 >= 0, s45 <= 1 + s41 + s42 + (1 + s43 + s44) + 1, z' = 1 + x_19 + x_29, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encode_f(z, z') -{ s49 + s50 + 3*x_110 + 4*x_110^2 + 3*x_17 + 4*x_17^2 + 3*x_210 + 4*x_210^2 + 3*x_27 + 4*x_27^2 }-> s51 :|: s46 >= 0, s46 <= x_17 + 1, s47 >= 0, s47 <= x_27 + 1, s48 >= 0, s48 <= x_110 + 1, s49 >= 0, s49 <= x_210 + 1, s50 >= 0, s50 <= s48 + s49 + 1, s51 >= 0, s51 <= 1 + s46 + s47 + s50 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, x_110 >= 0, z' = 1 + x_110 + x_210, x_210 >= 0
   encode_f(z, z') -{ 3*x_17 + 4*x_17^2 + 3*x_27 + 4*x_27^2 }-> s54 :|: s52 >= 0, s52 <= x_17 + 1, s53 >= 0, s53 <= x_27 + 1, s54 >= 0, s54 <= 1 + s52 + s53 + 0 + 1, x_17 >= 0, z = 1 + x_17 + x_27, x_27 >= 0, z' >= 0
   encode_f(z, z') -{ 1 + s56 + s58 + s59 + 3*x_111 + 4*x_111^2 + 3*x_18 + 4*x_18^2 + 3*x_211 + 4*x_211^2 + 3*x_28 + 4*x_28^2 }-> s60 :|: s55 >= 0, s55 <= x_18 + 1, s56 >= 0, s56 <= x_28 + 1, s57 >= 0, s57 <= s55 + s56 + 1, s58 >= 0, s58 <= x_111 + 1, s59 >= 0, s59 <= x_211 + 1, s60 >= 0, s60 <= s57 + (1 + s58 + s59) + 1, z' = 1 + x_111 + x_211, x_18 >= 0, x_28 >= 0, x_211 >= 0, z = 1 + x_18 + x_28, x_111 >= 0
   encode_f(z, z') -{ s62 + s65 + s66 + 3*x_112 + 4*x_112^2 + 3*x_18 + 4*x_18^2 + 3*x_212 + 4*x_212^2 + 3*x_28 + 4*x_28^2 }-> s67 :|: s61 >= 0, s61 <= x_18 + 1, s62 >= 0, s62 <= x_28 + 1, s63 >= 0, s63 <= s61 + s62 + 1, s64 >= 0, s64 <= x_112 + 1, s65 >= 0, s65 <= x_212 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= s63 + s66 + 1, x_212 >= 0, z' = 1 + x_112 + x_212, x_18 >= 0, x_28 >= 0, x_112 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ s69 + 3*x_18 + 4*x_18^2 + 3*x_28 + 4*x_28^2 }-> s71 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= x_28 + 1, s70 >= 0, s70 <= s68 + s69 + 1, s71 >= 0, s71 <= s70 + 0 + 1, z' >= 0, x_18 >= 0, x_28 >= 0, z = 1 + x_18 + x_28
   encode_f(z, z') -{ 1 + s72 + s73 + 3*x_113 + 4*x_113^2 + 3*x_213 + 4*x_213^2 }-> s74 :|: s72 >= 0, s72 <= x_113 + 1, s73 >= 0, s73 <= x_213 + 1, s74 >= 0, s74 <= 0 + (1 + s72 + s73) + 1, z >= 0, z' = 1 + x_113 + x_213, x_113 >= 0, x_213 >= 0
   encode_f(z, z') -{ s76 + s77 + 3*x_114 + 4*x_114^2 + 3*x_214 + 4*x_214^2 }-> s78 :|: s75 >= 0, s75 <= x_114 + 1, s76 >= 0, s76 <= x_214 + 1, s77 >= 0, s77 <= s75 + s76 + 1, s78 >= 0, s78 <= 0 + s77 + 1, z >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_114 + x_214
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z, z') -{ 3*z + 4*z^2 + 3*z' + 4*z'^2 }-> 1 + s79 + s80 :|: s79 >= 0, s79 <= z + 1, s80 >= 0, s80 <= z' + 1, z >= 0, z' >= 0
   f(z, z') -{ 1 + y }-> s :|: s >= 0, s <= 1 + y + x + y + 1, z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  f: runtime: O(n^1) [z'], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^2) [3*z + 4*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [18 + 39*z + 48*z^2 + 51*z' + 48*z'^2], size: O(n^1) [5 + z + z']
  encode_g: runtime: O(n^2) [3*z + 4*z^2 + 3*z' + 4*z'^2], size: O(n^1) [3 + z + z']

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(47) FinalProof (FINISHED)
Computed overall runtime complexity
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(48)
BOUNDS(1, n^2)