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


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WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 95 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) CpxWeightedTrsRenamingProof [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), 0 ms]
(18) CpxRNTS
(19) InliningProof [UPPER BOUND(ID), 244 ms]
(20) CpxRNTS
(21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxRNTS
(23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 5 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 297 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 35 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 758 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 161 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 3431 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 2366 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 881 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 453 ms]
(48) CpxRNTS
(49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(50) CpxRNTS
(51) IntTrsBoundProof [UPPER BOUND(ID), 915 ms]
(52) CpxRNTS
(53) IntTrsBoundProof [UPPER BOUND(ID), 622 ms]
(54) CpxRNTS
(55) FinalProof [FINISHED, 0 ms]
(56) 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:

   +(-(x, y), z) -> -(+(x, z), y)
   -(+(x, y), y) -> 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_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))


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

(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:

   +(-(x, y), z) -> -(+(x, z), y)
   -(+(x, y), y) -> x

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

   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))

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:

   +(-(x, y), z) -> -(+(x, z), y)
   -(+(x, y), y) -> x

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

   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))

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:

   -(c_+(x, y), y) -> x
   +(c_-(x, y), z) -> -(+(x, z), y)

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

   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   +(x0, x1) -> c_+(x0, x1)
   -(x0, x1) -> c_-(x0, x1)

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:

   -(c_+(x, y), y) -> x [1]
   +(c_-(x, y), z) -> -(+(x, z), y) [1]
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) [0]
   +(x0, x1) -> c_+(x0, x1) [0]
   -(x0, x1) -> c_-(x0, x1) [0]

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

(9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus
+ => plus

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

(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:

   minus(c_+(x, y), y) -> x [1]
   plus(c_-(x, y), z) -> minus(plus(x, z), y) [1]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_-(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   minus(x0, x1) -> c_-(x0, x1) [0]

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

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

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

   minus(c_+(x, y), y) -> x [1]
   plus(c_-(x, y), z) -> minus(plus(x, z), y) [1]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_-(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   minus(x0, x1) -> c_-(x0, x1) [0]

The TRS has the following type information:
minus :: c_+:c_- -> c_+:c_- -> c_+:c_-
c_+ :: c_+:c_- -> c_+:c_- -> c_+:c_-
plus :: c_+:c_- -> c_+:c_- -> c_+:c_-
c_- :: c_+:c_- -> c_+:c_- -> c_+:c_-
encArg :: cons_+:cons_- -> c_+:c_-
cons_+ :: cons_+:cons_- -> cons_+:cons_- -> cons_+:cons_-
cons_- :: cons_+:cons_- -> cons_+:cons_- -> cons_+:cons_-
encode_+ :: cons_+:cons_- -> cons_+:cons_- -> c_+:c_-
encode_- :: cons_+:cons_- -> cons_+:cons_- -> c_+:c_-

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_+_2
   encode_-_2
   plus_2
   minus_2

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_+(v0, v1) -> const [0]
   encode_-(v0, v1) -> const [0]
   plus(v0, v1) -> const [0]
   minus(v0, v1) -> const [0]

And the following fresh constants:    const, const1

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

(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:

   minus(c_+(x, y), y) -> x [1]
   plus(c_-(x, y), z) -> minus(plus(x, z), y) [1]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_-(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   minus(x0, x1) -> c_-(x0, x1) [0]
   encArg(v0) -> const [0]
   encode_+(v0, v1) -> const [0]
   encode_-(v0, v1) -> const [0]
   plus(v0, v1) -> const [0]
   minus(v0, v1) -> const [0]

The TRS has the following type information:
minus :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
c_+ :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
plus :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
c_- :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
encArg :: cons_+:cons_- -> c_+:c_-:const
cons_+ :: cons_+:cons_- -> cons_+:cons_- -> cons_+:cons_-
cons_- :: cons_+:cons_- -> cons_+:cons_- -> cons_+:cons_-
encode_+ :: cons_+:cons_- -> cons_+:cons_- -> c_+:c_-:const
encode_- :: cons_+:cons_- -> cons_+:cons_- -> c_+:c_-:const
const :: c_+:c_-:const
const1 :: cons_+:cons_-

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

(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:

   minus(c_+(x, y), y) -> x [1]
   plus(c_-(c_-(x', y'), y), z) -> minus(minus(plus(x', z), y'), y) [2]
   plus(c_-(x, y), z) -> minus(c_+(x, z), y) [1]
   plus(c_-(x, y), z) -> minus(const, y) [1]
   encArg(cons_+(cons_+(x_1', x_2'), cons_+(x_11, x_21))) -> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) [0]
   encArg(cons_+(cons_+(x_1', x_2'), cons_-(x_12, x_22))) -> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) [0]
   encArg(cons_+(cons_+(x_1', x_2'), x_2)) -> plus(plus(encArg(x_1'), encArg(x_2')), const) [0]
   encArg(cons_+(cons_-(x_1'', x_2''), cons_+(x_13, x_23))) -> plus(minus(encArg(x_1''), encArg(x_2'')), plus(encArg(x_13), encArg(x_23))) [0]
   encArg(cons_+(cons_-(x_1'', x_2''), cons_-(x_14, x_24))) -> plus(minus(encArg(x_1''), encArg(x_2'')), minus(encArg(x_14), encArg(x_24))) [0]
   encArg(cons_+(cons_-(x_1'', x_2''), x_2)) -> plus(minus(encArg(x_1''), encArg(x_2'')), const) [0]
   encArg(cons_+(x_1, cons_+(x_15, x_25))) -> plus(const, plus(encArg(x_15), encArg(x_25))) [0]
   encArg(cons_+(x_1, cons_-(x_16, x_26))) -> plus(const, minus(encArg(x_16), encArg(x_26))) [0]
   encArg(cons_+(x_1, x_2)) -> plus(const, const) [0]
   encArg(cons_-(cons_+(x_17, x_27), cons_+(x_19, x_29))) -> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) [0]
   encArg(cons_-(cons_+(x_17, x_27), cons_-(x_110, x_210))) -> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) [0]
   encArg(cons_-(cons_+(x_17, x_27), x_2)) -> minus(plus(encArg(x_17), encArg(x_27)), const) [0]
   encArg(cons_-(cons_-(x_18, x_28), cons_+(x_111, x_211))) -> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) [0]
   encArg(cons_-(cons_-(x_18, x_28), cons_-(x_112, x_212))) -> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) [0]
   encArg(cons_-(cons_-(x_18, x_28), x_2)) -> minus(minus(encArg(x_18), encArg(x_28)), const) [0]
   encArg(cons_-(x_1, cons_+(x_113, x_213))) -> minus(const, plus(encArg(x_113), encArg(x_213))) [0]
   encArg(cons_-(x_1, cons_-(x_114, x_214))) -> minus(const, minus(encArg(x_114), encArg(x_214))) [0]
   encArg(cons_-(x_1, x_2)) -> minus(const, const) [0]
   encode_+(cons_+(x_115, x_215), cons_+(x_117, x_217)) -> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) [0]
   encode_+(cons_+(x_115, x_215), cons_-(x_118, x_218)) -> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) [0]
   encode_+(cons_+(x_115, x_215), x_2) -> plus(plus(encArg(x_115), encArg(x_215)), const) [0]
   encode_+(cons_-(x_116, x_216), cons_+(x_119, x_219)) -> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) [0]
   encode_+(cons_-(x_116, x_216), cons_-(x_120, x_220)) -> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) [0]
   encode_+(cons_-(x_116, x_216), x_2) -> plus(minus(encArg(x_116), encArg(x_216)), const) [0]
   encode_+(x_1, cons_+(x_121, x_221)) -> plus(const, plus(encArg(x_121), encArg(x_221))) [0]
   encode_+(x_1, cons_-(x_122, x_222)) -> plus(const, minus(encArg(x_122), encArg(x_222))) [0]
   encode_+(x_1, x_2) -> plus(const, const) [0]
   encode_-(cons_+(x_123, x_223), cons_+(x_125, x_225)) -> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) [0]
   encode_-(cons_+(x_123, x_223), cons_-(x_126, x_226)) -> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) [0]
   encode_-(cons_+(x_123, x_223), x_2) -> minus(plus(encArg(x_123), encArg(x_223)), const) [0]
   encode_-(cons_-(x_124, x_224), cons_+(x_127, x_227)) -> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) [0]
   encode_-(cons_-(x_124, x_224), cons_-(x_128, x_228)) -> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) [0]
   encode_-(cons_-(x_124, x_224), x_2) -> minus(minus(encArg(x_124), encArg(x_224)), const) [0]
   encode_-(x_1, cons_+(x_129, x_229)) -> minus(const, plus(encArg(x_129), encArg(x_229))) [0]
   encode_-(x_1, cons_-(x_130, x_230)) -> minus(const, minus(encArg(x_130), encArg(x_230))) [0]
   encode_-(x_1, x_2) -> minus(const, const) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   minus(x0, x1) -> c_-(x0, x1) [0]
   encArg(v0) -> const [0]
   encode_+(v0, v1) -> const [0]
   encode_-(v0, v1) -> const [0]
   plus(v0, v1) -> const [0]
   minus(v0, v1) -> const [0]

The TRS has the following type information:
minus :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
c_+ :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
plus :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
c_- :: c_+:c_-:const -> c_+:c_-:const -> c_+:c_-:const
encArg :: cons_+:cons_- -> c_+:c_-:const
cons_+ :: cons_+:cons_- -> cons_+:cons_- -> cons_+:cons_-
cons_- :: cons_+:cons_- -> cons_+:cons_- -> cons_+:cons_-
encode_+ :: cons_+:cons_- -> cons_+:cons_- -> c_+:c_-:const
encode_- :: cons_+:cons_- -> cons_+:cons_- -> c_+:c_-:const
const :: c_+:c_-:const
const1 :: cons_+:cons_-

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

(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
   const1 => 0

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> minus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_2 >= 0, z'' = x_2
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, x_2 >= 0, z'' = x_2
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: x_1 >= 0, z' = x_1, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: x_1 >= 0, x_222 >= 0, z' = x_1, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_+(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, x_2 >= 0, z' = 1 + x_123 + x_223, z'' = x_2
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, x_2 >= 0, z'' = x_2
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: x_1 >= 0, z' = x_1, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: x_1 >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, z' = x_1, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, 0) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_-(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   minus(z', z'') -{ 1 }-> x :|: z'' = y, z' = 1 + x + y, x >= 0, y >= 0
   minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   minus(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z), y'), y) :|: z'' = z, z >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 1 }-> minus(0, y) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   plus(z', z'') -{ 1 }-> minus(1 + x + z, y) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0


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

(19) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   minus(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0
   minus(z', z'') -{ 1 }-> x :|: z'' = y, z' = 1 + x + y, x >= 0, y >= 0
   minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_2 >= 0, z'' = x_2
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, x_2 >= 0, z'' = x_2
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: x_1 >= 0, z' = x_1, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: x_1 >= 0, x_222 >= 0, z' = x_1, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_+(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, x_2 >= 0, z' = 1 + x_123 + x_223, z'' = x_2
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, x_2 >= 0, z'' = x_2
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: x_1 >= 0, z' = x_1, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: x_1 >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, z' = x_1, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_-(z', z'') -{ 0 }-> 0 :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z'' = y, z' = 1 + x + y, x >= 0, y >= 0
   minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   minus(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0
   plus(z', z'') -{ 2 }-> x' :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z), y'), y) :|: z'' = z, z >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0


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

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0


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

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

   { minus }
   { plus }
   { encArg }
   { encode_- }
   { encode_+ }

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {minus}, {plus}, {encArg}, {encode_-}, {encode_+}

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

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {minus}, {plus}, {encArg}, {encode_-}, {encode_+}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {plus}, {encArg}, {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']

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

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {plus}, {encArg}, {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6 + 4*z'

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

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

   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> plus(0, 0) :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 2 }-> minus(minus(plus(x', z''), y'), y) :|: z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']

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

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

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

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']

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

(39) 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'

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

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

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encArg}, {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: ?, size: O(n^1) [1 + z']

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

(41) 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: 6 + 21*z' + 8*z'^2

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

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

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), minus(encArg(x_12), encArg(x_22))) :|: x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(minus(encArg(x_1''), encArg(x_2'')), plus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), minus(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 }-> plus(minus(encArg(x_1''), encArg(x_2'')), 0) :|: z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> plus(0, plus(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 }-> plus(0, minus(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 }-> minus(plus(encArg(x_17), encArg(x_27)), plus(encArg(x_19), encArg(x_29))) :|: z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), minus(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 0 }-> minus(plus(encArg(x_17), encArg(x_27)), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), plus(encArg(x_111), encArg(x_211))) :|: z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), minus(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 0 }-> minus(minus(encArg(x_18), encArg(x_28)), 0) :|: x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> minus(0, plus(encArg(x_113), encArg(x_213))) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 0 }-> minus(0, minus(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), plus(encArg(x_117), encArg(x_217))) :|: x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), minus(encArg(x_118), encArg(x_218))) :|: z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 0 }-> plus(plus(encArg(x_115), encArg(x_215)), 0) :|: z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), plus(encArg(x_119), encArg(x_219))) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), minus(encArg(x_120), encArg(x_220))) :|: x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 0 }-> plus(minus(encArg(x_116), encArg(x_216)), 0) :|: x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, plus(encArg(x_121), encArg(x_221))) :|: z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> plus(0, minus(encArg(x_122), encArg(x_222))) :|: z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), plus(encArg(x_125), encArg(x_225))) :|: x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), minus(encArg(x_126), encArg(x_226))) :|: x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 0 }-> minus(plus(encArg(x_123), encArg(x_223)), 0) :|: x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), plus(encArg(x_127), encArg(x_227))) :|: x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), minus(encArg(x_128), encArg(x_228))) :|: x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 0 }-> minus(minus(encArg(x_124), encArg(x_224)), 0) :|: x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, plus(encArg(x_129), encArg(x_229))) :|: z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 0 }-> minus(0, minus(encArg(x_130), encArg(x_230))) :|: z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [6 + 21*z' + 8*z'^2], size: O(n^1) [1 + z']

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

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

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 19 + 4*s98 + 21*x_113 + 8*x_113^2 + 21*x_213 + 8*x_213^2 }-> s101 :|: s98 >= 0, s98 <= x_113 + 1, s99 >= 0, s99 <= x_213 + 1, s100 >= 0, s100 <= s98 + s99 + 1, s101 >= 0, s101 <= 0 + s100 + 1, x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 14 + 21*x_114 + 8*x_114^2 + 21*x_214 + 8*x_214^2 }-> s105 :|: s102 >= 0, s102 <= x_114 + 1, s103 >= 0, s103 <= x_214 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= 0 + s104 + 1, x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 24 + 4*s10 + 4*s12 + 21*x_1' + 8*x_1'^2 + 21*x_2' + 8*x_2'^2 }-> s13 :|: s10 >= 0, s10 <= x_1' + 1, s11 >= 0, s11 <= x_2' + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= s12 + 0 + 1, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 24 + 4*s14 + 21*x_15 + 8*x_15^2 + 21*x_25 + 8*x_25^2 }-> s17 :|: s14 >= 0, s14 <= x_15 + 1, s15 >= 0, s15 <= x_25 + 1, s16 >= 0, s16 <= s14 + s15 + 1, s17 >= 0, s17 <= 0 + s16 + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 37 + 4*s33 + 4*s35 + 21*x_1' + 8*x_1'^2 + 21*x_12 + 8*x_12^2 + 21*x_2' + 8*x_2'^2 + 21*x_22 + 8*x_22^2 }-> s39 :|: s33 >= 0, s33 <= x_1' + 1, s34 >= 0, s34 <= x_2' + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= x_12 + 1, s37 >= 0, s37 <= x_22 + 1, s38 >= 0, s38 <= s36 + s37 + 1, s39 >= 0, s39 <= s35 + s38 + 1, x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 37 + 4*s42 + 4*s43 + 21*x_1'' + 8*x_1''^2 + 21*x_13 + 8*x_13^2 + 21*x_2'' + 8*x_2''^2 + 21*x_23 + 8*x_23^2 }-> s46 :|: s40 >= 0, s40 <= x_1'' + 1, s41 >= 0, s41 <= x_2'' + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= x_13 + 1, s44 >= 0, s44 <= x_23 + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s42 + s45 + 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') -{ 32 + 4*s49 + 21*x_1'' + 8*x_1''^2 + 21*x_14 + 8*x_14^2 + 21*x_2'' + 8*x_2''^2 + 21*x_24 + 8*x_24^2 }-> s53 :|: s47 >= 0, s47 <= x_1'' + 1, s48 >= 0, s48 <= x_2'' + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= x_14 + 1, s51 >= 0, s51 <= x_24 + 1, s52 >= 0, s52 <= s50 + s51 + 1, s53 >= 0, s53 <= s49 + s52 + 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') -{ 19 + 4*s56 + 21*x_1'' + 8*x_1''^2 + 21*x_2'' + 8*x_2''^2 }-> s57 :|: s54 >= 0, s54 <= x_1'' + 1, s55 >= 0, s55 <= x_2'' + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 0 + 1, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 19 + 21*x_16 + 8*x_16^2 + 21*x_26 + 8*x_26^2 }-> s61 :|: s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= x_26 + 1, s60 >= 0, s60 <= s58 + s59 + 1, s61 >= 0, s61 <= 0 + s60 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 37 + 4*s62 + 4*s65 + 21*x_17 + 8*x_17^2 + 21*x_19 + 8*x_19^2 + 21*x_27 + 8*x_27^2 + 21*x_29 + 8*x_29^2 }-> s68 :|: s62 >= 0, s62 <= x_17 + 1, s63 >= 0, s63 <= x_27 + 1, s64 >= 0, s64 <= s62 + s63 + 1, s65 >= 0, s65 <= x_19 + 1, s66 >= 0, s66 <= x_29 + 1, s67 >= 0, s67 <= s65 + s66 + 1, s68 >= 0, s68 <= s64 + s67 + 1, z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 32 + 4*s69 + 21*x_110 + 8*x_110^2 + 21*x_17 + 8*x_17^2 + 21*x_210 + 8*x_210^2 + 21*x_27 + 8*x_27^2 }-> s75 :|: s69 >= 0, s69 <= x_17 + 1, s70 >= 0, s70 <= x_27 + 1, s71 >= 0, s71 <= s69 + s70 + 1, s72 >= 0, s72 <= x_110 + 1, s73 >= 0, s73 <= x_210 + 1, s74 >= 0, s74 <= s72 + s73 + 1, s75 >= 0, s75 <= s71 + s74 + 1, x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 19 + 4*s76 + 21*x_17 + 8*x_17^2 + 21*x_27 + 8*x_27^2 }-> s79 :|: s76 >= 0, s76 <= x_17 + 1, s77 >= 0, s77 <= x_27 + 1, s78 >= 0, s78 <= s76 + s77 + 1, s79 >= 0, s79 <= s78 + 0 + 1, x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 32 + 4*s83 + 21*x_111 + 8*x_111^2 + 21*x_18 + 8*x_18^2 + 21*x_211 + 8*x_211^2 + 21*x_28 + 8*x_28^2 }-> s86 :|: s80 >= 0, s80 <= x_18 + 1, s81 >= 0, s81 <= x_28 + 1, s82 >= 0, s82 <= s80 + s81 + 1, s83 >= 0, s83 <= x_111 + 1, s84 >= 0, s84 <= x_211 + 1, s85 >= 0, s85 <= s83 + s84 + 1, s86 >= 0, s86 <= s82 + s85 + 1, z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 42 + 4*s3 + 4*s5 + 4*s6 + 21*x_1' + 8*x_1'^2 + 21*x_11 + 8*x_11^2 + 21*x_2' + 8*x_2'^2 + 21*x_21 + 8*x_21^2 }-> s9 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= s3 + s4 + 1, s6 >= 0, s6 <= x_11 + 1, s7 >= 0, s7 <= x_21 + 1, s8 >= 0, s8 <= s6 + s7 + 1, s9 >= 0, s9 <= s5 + s8 + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 27 + 21*x_112 + 8*x_112^2 + 21*x_18 + 8*x_18^2 + 21*x_212 + 8*x_212^2 + 21*x_28 + 8*x_28^2 }-> s93 :|: s87 >= 0, s87 <= x_18 + 1, s88 >= 0, s88 <= x_28 + 1, s89 >= 0, s89 <= s87 + s88 + 1, s90 >= 0, s90 <= x_112 + 1, s91 >= 0, s91 <= x_212 + 1, s92 >= 0, s92 <= s90 + s91 + 1, s93 >= 0, s93 <= s89 + s92 + 1, x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 14 + 21*x_18 + 8*x_18^2 + 21*x_28 + 8*x_28^2 }-> s97 :|: s94 >= 0, s94 <= x_18 + 1, s95 >= 0, s95 <= x_28 + 1, s96 >= 0, s96 <= s94 + s95 + 1, s97 >= 0, s97 <= s96 + 0 + 1, x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 37 + 4*s106 + 4*s108 + 21*x_115 + 8*x_115^2 + 21*x_118 + 8*x_118^2 + 21*x_215 + 8*x_215^2 + 21*x_218 + 8*x_218^2 }-> s112 :|: s106 >= 0, s106 <= x_115 + 1, s107 >= 0, s107 <= x_215 + 1, s108 >= 0, s108 <= s106 + s107 + 1, s109 >= 0, s109 <= x_118 + 1, s110 >= 0, s110 <= x_218 + 1, s111 >= 0, s111 <= s109 + s110 + 1, s112 >= 0, s112 <= s108 + s111 + 1, z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 37 + 4*s115 + 4*s116 + 21*x_116 + 8*x_116^2 + 21*x_119 + 8*x_119^2 + 21*x_216 + 8*x_216^2 + 21*x_219 + 8*x_219^2 }-> s119 :|: s113 >= 0, s113 <= x_116 + 1, s114 >= 0, s114 <= x_216 + 1, s115 >= 0, s115 <= s113 + s114 + 1, s116 >= 0, s116 <= x_119 + 1, s117 >= 0, s117 <= x_219 + 1, s118 >= 0, s118 <= s116 + s117 + 1, s119 >= 0, s119 <= s115 + s118 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 32 + 4*s122 + 21*x_116 + 8*x_116^2 + 21*x_120 + 8*x_120^2 + 21*x_216 + 8*x_216^2 + 21*x_220 + 8*x_220^2 }-> s126 :|: s120 >= 0, s120 <= x_116 + 1, s121 >= 0, s121 <= x_216 + 1, s122 >= 0, s122 <= s120 + s121 + 1, s123 >= 0, s123 <= x_120 + 1, s124 >= 0, s124 <= x_220 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= s122 + s125 + 1, x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 19 + 4*s129 + 21*x_116 + 8*x_116^2 + 21*x_216 + 8*x_216^2 }-> s130 :|: s127 >= 0, s127 <= x_116 + 1, s128 >= 0, s128 <= x_216 + 1, s129 >= 0, s129 <= s127 + s128 + 1, s130 >= 0, s130 <= s129 + 0 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 19 + 21*x_122 + 8*x_122^2 + 21*x_222 + 8*x_222^2 }-> s134 :|: s131 >= 0, s131 <= x_122 + 1, s132 >= 0, s132 <= x_222 + 1, s133 >= 0, s133 <= s131 + s132 + 1, s134 >= 0, s134 <= 0 + s133 + 1, z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 42 + 4*s18 + 4*s20 + 4*s21 + 21*x_115 + 8*x_115^2 + 21*x_117 + 8*x_117^2 + 21*x_215 + 8*x_215^2 + 21*x_217 + 8*x_217^2 }-> s24 :|: s18 >= 0, s18 <= x_115 + 1, s19 >= 0, s19 <= x_215 + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= x_117 + 1, s22 >= 0, s22 <= x_217 + 1, s23 >= 0, s23 <= s21 + s22 + 1, s24 >= 0, s24 <= s20 + s23 + 1, x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 24 + 4*s25 + 4*s27 + 21*x_115 + 8*x_115^2 + 21*x_215 + 8*x_215^2 }-> s28 :|: s25 >= 0, s25 <= x_115 + 1, s26 >= 0, s26 <= x_215 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 0 + 1, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 24 + 4*s29 + 21*x_121 + 8*x_121^2 + 21*x_221 + 8*x_221^2 }-> s32 :|: s29 >= 0, s29 <= x_121 + 1, s30 >= 0, s30 <= x_221 + 1, s31 >= 0, s31 <= s29 + s30 + 1, s32 >= 0, s32 <= 0 + s31 + 1, z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 37 + 4*s135 + 4*s138 + 21*x_123 + 8*x_123^2 + 21*x_125 + 8*x_125^2 + 21*x_223 + 8*x_223^2 + 21*x_225 + 8*x_225^2 }-> s141 :|: s135 >= 0, s135 <= x_123 + 1, s136 >= 0, s136 <= x_223 + 1, s137 >= 0, s137 <= s135 + s136 + 1, s138 >= 0, s138 <= x_125 + 1, s139 >= 0, s139 <= x_225 + 1, s140 >= 0, s140 <= s138 + s139 + 1, s141 >= 0, s141 <= s137 + s140 + 1, x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s142 + 21*x_123 + 8*x_123^2 + 21*x_126 + 8*x_126^2 + 21*x_223 + 8*x_223^2 + 21*x_226 + 8*x_226^2 }-> s148 :|: s142 >= 0, s142 <= x_123 + 1, s143 >= 0, s143 <= x_223 + 1, s144 >= 0, s144 <= s142 + s143 + 1, s145 >= 0, s145 <= x_126 + 1, s146 >= 0, s146 <= x_226 + 1, s147 >= 0, s147 <= s145 + s146 + 1, s148 >= 0, s148 <= s144 + s147 + 1, x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 19 + 4*s149 + 21*x_123 + 8*x_123^2 + 21*x_223 + 8*x_223^2 }-> s152 :|: s149 >= 0, s149 <= x_123 + 1, s150 >= 0, s150 <= x_223 + 1, s151 >= 0, s151 <= s149 + s150 + 1, s152 >= 0, s152 <= s151 + 0 + 1, x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s156 + 21*x_124 + 8*x_124^2 + 21*x_127 + 8*x_127^2 + 21*x_224 + 8*x_224^2 + 21*x_227 + 8*x_227^2 }-> s159 :|: s153 >= 0, s153 <= x_124 + 1, s154 >= 0, s154 <= x_224 + 1, s155 >= 0, s155 <= s153 + s154 + 1, s156 >= 0, s156 <= x_127 + 1, s157 >= 0, s157 <= x_227 + 1, s158 >= 0, s158 <= s156 + s157 + 1, s159 >= 0, s159 <= s155 + s158 + 1, x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 27 + 21*x_124 + 8*x_124^2 + 21*x_128 + 8*x_128^2 + 21*x_224 + 8*x_224^2 + 21*x_228 + 8*x_228^2 }-> s166 :|: s160 >= 0, s160 <= x_124 + 1, s161 >= 0, s161 <= x_224 + 1, s162 >= 0, s162 <= s160 + s161 + 1, s163 >= 0, s163 <= x_128 + 1, s164 >= 0, s164 <= x_228 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s162 + s165 + 1, x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 14 + 21*x_124 + 8*x_124^2 + 21*x_224 + 8*x_224^2 }-> s170 :|: s167 >= 0, s167 <= x_124 + 1, s168 >= 0, s168 <= x_224 + 1, s169 >= 0, s169 <= s167 + s168 + 1, s170 >= 0, s170 <= s169 + 0 + 1, x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 19 + 4*s171 + 21*x_129 + 8*x_129^2 + 21*x_229 + 8*x_229^2 }-> s174 :|: s171 >= 0, s171 <= x_129 + 1, s172 >= 0, s172 <= x_229 + 1, s173 >= 0, s173 <= s171 + s172 + 1, s174 >= 0, s174 <= 0 + s173 + 1, z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 14 + 21*x_130 + 8*x_130^2 + 21*x_230 + 8*x_230^2 }-> s178 :|: s175 >= 0, s175 <= x_130 + 1, s176 >= 0, s176 <= x_230 + 1, s177 >= 0, s177 <= s175 + s176 + 1, s178 >= 0, s178 <= 0 + s177 + 1, z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [6 + 21*z' + 8*z'^2], size: O(n^1) [1 + z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 19 + 4*s98 + 21*x_113 + 8*x_113^2 + 21*x_213 + 8*x_213^2 }-> s101 :|: s98 >= 0, s98 <= x_113 + 1, s99 >= 0, s99 <= x_213 + 1, s100 >= 0, s100 <= s98 + s99 + 1, s101 >= 0, s101 <= 0 + s100 + 1, x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 14 + 21*x_114 + 8*x_114^2 + 21*x_214 + 8*x_214^2 }-> s105 :|: s102 >= 0, s102 <= x_114 + 1, s103 >= 0, s103 <= x_214 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= 0 + s104 + 1, x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 24 + 4*s10 + 4*s12 + 21*x_1' + 8*x_1'^2 + 21*x_2' + 8*x_2'^2 }-> s13 :|: s10 >= 0, s10 <= x_1' + 1, s11 >= 0, s11 <= x_2' + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= s12 + 0 + 1, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 24 + 4*s14 + 21*x_15 + 8*x_15^2 + 21*x_25 + 8*x_25^2 }-> s17 :|: s14 >= 0, s14 <= x_15 + 1, s15 >= 0, s15 <= x_25 + 1, s16 >= 0, s16 <= s14 + s15 + 1, s17 >= 0, s17 <= 0 + s16 + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 37 + 4*s33 + 4*s35 + 21*x_1' + 8*x_1'^2 + 21*x_12 + 8*x_12^2 + 21*x_2' + 8*x_2'^2 + 21*x_22 + 8*x_22^2 }-> s39 :|: s33 >= 0, s33 <= x_1' + 1, s34 >= 0, s34 <= x_2' + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= x_12 + 1, s37 >= 0, s37 <= x_22 + 1, s38 >= 0, s38 <= s36 + s37 + 1, s39 >= 0, s39 <= s35 + s38 + 1, x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 37 + 4*s42 + 4*s43 + 21*x_1'' + 8*x_1''^2 + 21*x_13 + 8*x_13^2 + 21*x_2'' + 8*x_2''^2 + 21*x_23 + 8*x_23^2 }-> s46 :|: s40 >= 0, s40 <= x_1'' + 1, s41 >= 0, s41 <= x_2'' + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= x_13 + 1, s44 >= 0, s44 <= x_23 + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s42 + s45 + 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') -{ 32 + 4*s49 + 21*x_1'' + 8*x_1''^2 + 21*x_14 + 8*x_14^2 + 21*x_2'' + 8*x_2''^2 + 21*x_24 + 8*x_24^2 }-> s53 :|: s47 >= 0, s47 <= x_1'' + 1, s48 >= 0, s48 <= x_2'' + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= x_14 + 1, s51 >= 0, s51 <= x_24 + 1, s52 >= 0, s52 <= s50 + s51 + 1, s53 >= 0, s53 <= s49 + s52 + 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') -{ 19 + 4*s56 + 21*x_1'' + 8*x_1''^2 + 21*x_2'' + 8*x_2''^2 }-> s57 :|: s54 >= 0, s54 <= x_1'' + 1, s55 >= 0, s55 <= x_2'' + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 0 + 1, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 19 + 21*x_16 + 8*x_16^2 + 21*x_26 + 8*x_26^2 }-> s61 :|: s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= x_26 + 1, s60 >= 0, s60 <= s58 + s59 + 1, s61 >= 0, s61 <= 0 + s60 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 37 + 4*s62 + 4*s65 + 21*x_17 + 8*x_17^2 + 21*x_19 + 8*x_19^2 + 21*x_27 + 8*x_27^2 + 21*x_29 + 8*x_29^2 }-> s68 :|: s62 >= 0, s62 <= x_17 + 1, s63 >= 0, s63 <= x_27 + 1, s64 >= 0, s64 <= s62 + s63 + 1, s65 >= 0, s65 <= x_19 + 1, s66 >= 0, s66 <= x_29 + 1, s67 >= 0, s67 <= s65 + s66 + 1, s68 >= 0, s68 <= s64 + s67 + 1, z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 32 + 4*s69 + 21*x_110 + 8*x_110^2 + 21*x_17 + 8*x_17^2 + 21*x_210 + 8*x_210^2 + 21*x_27 + 8*x_27^2 }-> s75 :|: s69 >= 0, s69 <= x_17 + 1, s70 >= 0, s70 <= x_27 + 1, s71 >= 0, s71 <= s69 + s70 + 1, s72 >= 0, s72 <= x_110 + 1, s73 >= 0, s73 <= x_210 + 1, s74 >= 0, s74 <= s72 + s73 + 1, s75 >= 0, s75 <= s71 + s74 + 1, x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 19 + 4*s76 + 21*x_17 + 8*x_17^2 + 21*x_27 + 8*x_27^2 }-> s79 :|: s76 >= 0, s76 <= x_17 + 1, s77 >= 0, s77 <= x_27 + 1, s78 >= 0, s78 <= s76 + s77 + 1, s79 >= 0, s79 <= s78 + 0 + 1, x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 32 + 4*s83 + 21*x_111 + 8*x_111^2 + 21*x_18 + 8*x_18^2 + 21*x_211 + 8*x_211^2 + 21*x_28 + 8*x_28^2 }-> s86 :|: s80 >= 0, s80 <= x_18 + 1, s81 >= 0, s81 <= x_28 + 1, s82 >= 0, s82 <= s80 + s81 + 1, s83 >= 0, s83 <= x_111 + 1, s84 >= 0, s84 <= x_211 + 1, s85 >= 0, s85 <= s83 + s84 + 1, s86 >= 0, s86 <= s82 + s85 + 1, z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 42 + 4*s3 + 4*s5 + 4*s6 + 21*x_1' + 8*x_1'^2 + 21*x_11 + 8*x_11^2 + 21*x_2' + 8*x_2'^2 + 21*x_21 + 8*x_21^2 }-> s9 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= s3 + s4 + 1, s6 >= 0, s6 <= x_11 + 1, s7 >= 0, s7 <= x_21 + 1, s8 >= 0, s8 <= s6 + s7 + 1, s9 >= 0, s9 <= s5 + s8 + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 27 + 21*x_112 + 8*x_112^2 + 21*x_18 + 8*x_18^2 + 21*x_212 + 8*x_212^2 + 21*x_28 + 8*x_28^2 }-> s93 :|: s87 >= 0, s87 <= x_18 + 1, s88 >= 0, s88 <= x_28 + 1, s89 >= 0, s89 <= s87 + s88 + 1, s90 >= 0, s90 <= x_112 + 1, s91 >= 0, s91 <= x_212 + 1, s92 >= 0, s92 <= s90 + s91 + 1, s93 >= 0, s93 <= s89 + s92 + 1, x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 14 + 21*x_18 + 8*x_18^2 + 21*x_28 + 8*x_28^2 }-> s97 :|: s94 >= 0, s94 <= x_18 + 1, s95 >= 0, s95 <= x_28 + 1, s96 >= 0, s96 <= s94 + s95 + 1, s97 >= 0, s97 <= s96 + 0 + 1, x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 37 + 4*s106 + 4*s108 + 21*x_115 + 8*x_115^2 + 21*x_118 + 8*x_118^2 + 21*x_215 + 8*x_215^2 + 21*x_218 + 8*x_218^2 }-> s112 :|: s106 >= 0, s106 <= x_115 + 1, s107 >= 0, s107 <= x_215 + 1, s108 >= 0, s108 <= s106 + s107 + 1, s109 >= 0, s109 <= x_118 + 1, s110 >= 0, s110 <= x_218 + 1, s111 >= 0, s111 <= s109 + s110 + 1, s112 >= 0, s112 <= s108 + s111 + 1, z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 37 + 4*s115 + 4*s116 + 21*x_116 + 8*x_116^2 + 21*x_119 + 8*x_119^2 + 21*x_216 + 8*x_216^2 + 21*x_219 + 8*x_219^2 }-> s119 :|: s113 >= 0, s113 <= x_116 + 1, s114 >= 0, s114 <= x_216 + 1, s115 >= 0, s115 <= s113 + s114 + 1, s116 >= 0, s116 <= x_119 + 1, s117 >= 0, s117 <= x_219 + 1, s118 >= 0, s118 <= s116 + s117 + 1, s119 >= 0, s119 <= s115 + s118 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 32 + 4*s122 + 21*x_116 + 8*x_116^2 + 21*x_120 + 8*x_120^2 + 21*x_216 + 8*x_216^2 + 21*x_220 + 8*x_220^2 }-> s126 :|: s120 >= 0, s120 <= x_116 + 1, s121 >= 0, s121 <= x_216 + 1, s122 >= 0, s122 <= s120 + s121 + 1, s123 >= 0, s123 <= x_120 + 1, s124 >= 0, s124 <= x_220 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= s122 + s125 + 1, x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 19 + 4*s129 + 21*x_116 + 8*x_116^2 + 21*x_216 + 8*x_216^2 }-> s130 :|: s127 >= 0, s127 <= x_116 + 1, s128 >= 0, s128 <= x_216 + 1, s129 >= 0, s129 <= s127 + s128 + 1, s130 >= 0, s130 <= s129 + 0 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 19 + 21*x_122 + 8*x_122^2 + 21*x_222 + 8*x_222^2 }-> s134 :|: s131 >= 0, s131 <= x_122 + 1, s132 >= 0, s132 <= x_222 + 1, s133 >= 0, s133 <= s131 + s132 + 1, s134 >= 0, s134 <= 0 + s133 + 1, z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 42 + 4*s18 + 4*s20 + 4*s21 + 21*x_115 + 8*x_115^2 + 21*x_117 + 8*x_117^2 + 21*x_215 + 8*x_215^2 + 21*x_217 + 8*x_217^2 }-> s24 :|: s18 >= 0, s18 <= x_115 + 1, s19 >= 0, s19 <= x_215 + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= x_117 + 1, s22 >= 0, s22 <= x_217 + 1, s23 >= 0, s23 <= s21 + s22 + 1, s24 >= 0, s24 <= s20 + s23 + 1, x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 24 + 4*s25 + 4*s27 + 21*x_115 + 8*x_115^2 + 21*x_215 + 8*x_215^2 }-> s28 :|: s25 >= 0, s25 <= x_115 + 1, s26 >= 0, s26 <= x_215 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 0 + 1, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 24 + 4*s29 + 21*x_121 + 8*x_121^2 + 21*x_221 + 8*x_221^2 }-> s32 :|: s29 >= 0, s29 <= x_121 + 1, s30 >= 0, s30 <= x_221 + 1, s31 >= 0, s31 <= s29 + s30 + 1, s32 >= 0, s32 <= 0 + s31 + 1, z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 37 + 4*s135 + 4*s138 + 21*x_123 + 8*x_123^2 + 21*x_125 + 8*x_125^2 + 21*x_223 + 8*x_223^2 + 21*x_225 + 8*x_225^2 }-> s141 :|: s135 >= 0, s135 <= x_123 + 1, s136 >= 0, s136 <= x_223 + 1, s137 >= 0, s137 <= s135 + s136 + 1, s138 >= 0, s138 <= x_125 + 1, s139 >= 0, s139 <= x_225 + 1, s140 >= 0, s140 <= s138 + s139 + 1, s141 >= 0, s141 <= s137 + s140 + 1, x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s142 + 21*x_123 + 8*x_123^2 + 21*x_126 + 8*x_126^2 + 21*x_223 + 8*x_223^2 + 21*x_226 + 8*x_226^2 }-> s148 :|: s142 >= 0, s142 <= x_123 + 1, s143 >= 0, s143 <= x_223 + 1, s144 >= 0, s144 <= s142 + s143 + 1, s145 >= 0, s145 <= x_126 + 1, s146 >= 0, s146 <= x_226 + 1, s147 >= 0, s147 <= s145 + s146 + 1, s148 >= 0, s148 <= s144 + s147 + 1, x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 19 + 4*s149 + 21*x_123 + 8*x_123^2 + 21*x_223 + 8*x_223^2 }-> s152 :|: s149 >= 0, s149 <= x_123 + 1, s150 >= 0, s150 <= x_223 + 1, s151 >= 0, s151 <= s149 + s150 + 1, s152 >= 0, s152 <= s151 + 0 + 1, x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s156 + 21*x_124 + 8*x_124^2 + 21*x_127 + 8*x_127^2 + 21*x_224 + 8*x_224^2 + 21*x_227 + 8*x_227^2 }-> s159 :|: s153 >= 0, s153 <= x_124 + 1, s154 >= 0, s154 <= x_224 + 1, s155 >= 0, s155 <= s153 + s154 + 1, s156 >= 0, s156 <= x_127 + 1, s157 >= 0, s157 <= x_227 + 1, s158 >= 0, s158 <= s156 + s157 + 1, s159 >= 0, s159 <= s155 + s158 + 1, x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 27 + 21*x_124 + 8*x_124^2 + 21*x_128 + 8*x_128^2 + 21*x_224 + 8*x_224^2 + 21*x_228 + 8*x_228^2 }-> s166 :|: s160 >= 0, s160 <= x_124 + 1, s161 >= 0, s161 <= x_224 + 1, s162 >= 0, s162 <= s160 + s161 + 1, s163 >= 0, s163 <= x_128 + 1, s164 >= 0, s164 <= x_228 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s162 + s165 + 1, x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 14 + 21*x_124 + 8*x_124^2 + 21*x_224 + 8*x_224^2 }-> s170 :|: s167 >= 0, s167 <= x_124 + 1, s168 >= 0, s168 <= x_224 + 1, s169 >= 0, s169 <= s167 + s168 + 1, s170 >= 0, s170 <= s169 + 0 + 1, x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 19 + 4*s171 + 21*x_129 + 8*x_129^2 + 21*x_229 + 8*x_229^2 }-> s174 :|: s171 >= 0, s171 <= x_129 + 1, s172 >= 0, s172 <= x_229 + 1, s173 >= 0, s173 <= s171 + s172 + 1, s174 >= 0, s174 <= 0 + s173 + 1, z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 14 + 21*x_130 + 8*x_130^2 + 21*x_230 + 8*x_230^2 }-> s178 :|: s175 >= 0, s175 <= x_130 + 1, s176 >= 0, s176 <= x_230 + 1, s177 >= 0, s177 <= s175 + s176 + 1, s178 >= 0, s178 <= 0 + s177 + 1, z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_-}, {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [6 + 21*z' + 8*z'^2], size: O(n^1) [1 + z']
  encode_-: runtime: ?, size: O(n^1) [5 + z' + z'']

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(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_-
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 218 + 264*z' + 96*z'^2 + 264*z'' + 96*z''^2

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

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

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 19 + 4*s98 + 21*x_113 + 8*x_113^2 + 21*x_213 + 8*x_213^2 }-> s101 :|: s98 >= 0, s98 <= x_113 + 1, s99 >= 0, s99 <= x_213 + 1, s100 >= 0, s100 <= s98 + s99 + 1, s101 >= 0, s101 <= 0 + s100 + 1, x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 14 + 21*x_114 + 8*x_114^2 + 21*x_214 + 8*x_214^2 }-> s105 :|: s102 >= 0, s102 <= x_114 + 1, s103 >= 0, s103 <= x_214 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= 0 + s104 + 1, x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 24 + 4*s10 + 4*s12 + 21*x_1' + 8*x_1'^2 + 21*x_2' + 8*x_2'^2 }-> s13 :|: s10 >= 0, s10 <= x_1' + 1, s11 >= 0, s11 <= x_2' + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= s12 + 0 + 1, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 24 + 4*s14 + 21*x_15 + 8*x_15^2 + 21*x_25 + 8*x_25^2 }-> s17 :|: s14 >= 0, s14 <= x_15 + 1, s15 >= 0, s15 <= x_25 + 1, s16 >= 0, s16 <= s14 + s15 + 1, s17 >= 0, s17 <= 0 + s16 + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 37 + 4*s33 + 4*s35 + 21*x_1' + 8*x_1'^2 + 21*x_12 + 8*x_12^2 + 21*x_2' + 8*x_2'^2 + 21*x_22 + 8*x_22^2 }-> s39 :|: s33 >= 0, s33 <= x_1' + 1, s34 >= 0, s34 <= x_2' + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= x_12 + 1, s37 >= 0, s37 <= x_22 + 1, s38 >= 0, s38 <= s36 + s37 + 1, s39 >= 0, s39 <= s35 + s38 + 1, x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 37 + 4*s42 + 4*s43 + 21*x_1'' + 8*x_1''^2 + 21*x_13 + 8*x_13^2 + 21*x_2'' + 8*x_2''^2 + 21*x_23 + 8*x_23^2 }-> s46 :|: s40 >= 0, s40 <= x_1'' + 1, s41 >= 0, s41 <= x_2'' + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= x_13 + 1, s44 >= 0, s44 <= x_23 + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s42 + s45 + 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') -{ 32 + 4*s49 + 21*x_1'' + 8*x_1''^2 + 21*x_14 + 8*x_14^2 + 21*x_2'' + 8*x_2''^2 + 21*x_24 + 8*x_24^2 }-> s53 :|: s47 >= 0, s47 <= x_1'' + 1, s48 >= 0, s48 <= x_2'' + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= x_14 + 1, s51 >= 0, s51 <= x_24 + 1, s52 >= 0, s52 <= s50 + s51 + 1, s53 >= 0, s53 <= s49 + s52 + 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') -{ 19 + 4*s56 + 21*x_1'' + 8*x_1''^2 + 21*x_2'' + 8*x_2''^2 }-> s57 :|: s54 >= 0, s54 <= x_1'' + 1, s55 >= 0, s55 <= x_2'' + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 0 + 1, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 19 + 21*x_16 + 8*x_16^2 + 21*x_26 + 8*x_26^2 }-> s61 :|: s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= x_26 + 1, s60 >= 0, s60 <= s58 + s59 + 1, s61 >= 0, s61 <= 0 + s60 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 37 + 4*s62 + 4*s65 + 21*x_17 + 8*x_17^2 + 21*x_19 + 8*x_19^2 + 21*x_27 + 8*x_27^2 + 21*x_29 + 8*x_29^2 }-> s68 :|: s62 >= 0, s62 <= x_17 + 1, s63 >= 0, s63 <= x_27 + 1, s64 >= 0, s64 <= s62 + s63 + 1, s65 >= 0, s65 <= x_19 + 1, s66 >= 0, s66 <= x_29 + 1, s67 >= 0, s67 <= s65 + s66 + 1, s68 >= 0, s68 <= s64 + s67 + 1, z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 32 + 4*s69 + 21*x_110 + 8*x_110^2 + 21*x_17 + 8*x_17^2 + 21*x_210 + 8*x_210^2 + 21*x_27 + 8*x_27^2 }-> s75 :|: s69 >= 0, s69 <= x_17 + 1, s70 >= 0, s70 <= x_27 + 1, s71 >= 0, s71 <= s69 + s70 + 1, s72 >= 0, s72 <= x_110 + 1, s73 >= 0, s73 <= x_210 + 1, s74 >= 0, s74 <= s72 + s73 + 1, s75 >= 0, s75 <= s71 + s74 + 1, x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 19 + 4*s76 + 21*x_17 + 8*x_17^2 + 21*x_27 + 8*x_27^2 }-> s79 :|: s76 >= 0, s76 <= x_17 + 1, s77 >= 0, s77 <= x_27 + 1, s78 >= 0, s78 <= s76 + s77 + 1, s79 >= 0, s79 <= s78 + 0 + 1, x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 32 + 4*s83 + 21*x_111 + 8*x_111^2 + 21*x_18 + 8*x_18^2 + 21*x_211 + 8*x_211^2 + 21*x_28 + 8*x_28^2 }-> s86 :|: s80 >= 0, s80 <= x_18 + 1, s81 >= 0, s81 <= x_28 + 1, s82 >= 0, s82 <= s80 + s81 + 1, s83 >= 0, s83 <= x_111 + 1, s84 >= 0, s84 <= x_211 + 1, s85 >= 0, s85 <= s83 + s84 + 1, s86 >= 0, s86 <= s82 + s85 + 1, z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 42 + 4*s3 + 4*s5 + 4*s6 + 21*x_1' + 8*x_1'^2 + 21*x_11 + 8*x_11^2 + 21*x_2' + 8*x_2'^2 + 21*x_21 + 8*x_21^2 }-> s9 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= s3 + s4 + 1, s6 >= 0, s6 <= x_11 + 1, s7 >= 0, s7 <= x_21 + 1, s8 >= 0, s8 <= s6 + s7 + 1, s9 >= 0, s9 <= s5 + s8 + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 27 + 21*x_112 + 8*x_112^2 + 21*x_18 + 8*x_18^2 + 21*x_212 + 8*x_212^2 + 21*x_28 + 8*x_28^2 }-> s93 :|: s87 >= 0, s87 <= x_18 + 1, s88 >= 0, s88 <= x_28 + 1, s89 >= 0, s89 <= s87 + s88 + 1, s90 >= 0, s90 <= x_112 + 1, s91 >= 0, s91 <= x_212 + 1, s92 >= 0, s92 <= s90 + s91 + 1, s93 >= 0, s93 <= s89 + s92 + 1, x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 14 + 21*x_18 + 8*x_18^2 + 21*x_28 + 8*x_28^2 }-> s97 :|: s94 >= 0, s94 <= x_18 + 1, s95 >= 0, s95 <= x_28 + 1, s96 >= 0, s96 <= s94 + s95 + 1, s97 >= 0, s97 <= s96 + 0 + 1, x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 37 + 4*s106 + 4*s108 + 21*x_115 + 8*x_115^2 + 21*x_118 + 8*x_118^2 + 21*x_215 + 8*x_215^2 + 21*x_218 + 8*x_218^2 }-> s112 :|: s106 >= 0, s106 <= x_115 + 1, s107 >= 0, s107 <= x_215 + 1, s108 >= 0, s108 <= s106 + s107 + 1, s109 >= 0, s109 <= x_118 + 1, s110 >= 0, s110 <= x_218 + 1, s111 >= 0, s111 <= s109 + s110 + 1, s112 >= 0, s112 <= s108 + s111 + 1, z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 37 + 4*s115 + 4*s116 + 21*x_116 + 8*x_116^2 + 21*x_119 + 8*x_119^2 + 21*x_216 + 8*x_216^2 + 21*x_219 + 8*x_219^2 }-> s119 :|: s113 >= 0, s113 <= x_116 + 1, s114 >= 0, s114 <= x_216 + 1, s115 >= 0, s115 <= s113 + s114 + 1, s116 >= 0, s116 <= x_119 + 1, s117 >= 0, s117 <= x_219 + 1, s118 >= 0, s118 <= s116 + s117 + 1, s119 >= 0, s119 <= s115 + s118 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 32 + 4*s122 + 21*x_116 + 8*x_116^2 + 21*x_120 + 8*x_120^2 + 21*x_216 + 8*x_216^2 + 21*x_220 + 8*x_220^2 }-> s126 :|: s120 >= 0, s120 <= x_116 + 1, s121 >= 0, s121 <= x_216 + 1, s122 >= 0, s122 <= s120 + s121 + 1, s123 >= 0, s123 <= x_120 + 1, s124 >= 0, s124 <= x_220 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= s122 + s125 + 1, x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 19 + 4*s129 + 21*x_116 + 8*x_116^2 + 21*x_216 + 8*x_216^2 }-> s130 :|: s127 >= 0, s127 <= x_116 + 1, s128 >= 0, s128 <= x_216 + 1, s129 >= 0, s129 <= s127 + s128 + 1, s130 >= 0, s130 <= s129 + 0 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 19 + 21*x_122 + 8*x_122^2 + 21*x_222 + 8*x_222^2 }-> s134 :|: s131 >= 0, s131 <= x_122 + 1, s132 >= 0, s132 <= x_222 + 1, s133 >= 0, s133 <= s131 + s132 + 1, s134 >= 0, s134 <= 0 + s133 + 1, z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 42 + 4*s18 + 4*s20 + 4*s21 + 21*x_115 + 8*x_115^2 + 21*x_117 + 8*x_117^2 + 21*x_215 + 8*x_215^2 + 21*x_217 + 8*x_217^2 }-> s24 :|: s18 >= 0, s18 <= x_115 + 1, s19 >= 0, s19 <= x_215 + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= x_117 + 1, s22 >= 0, s22 <= x_217 + 1, s23 >= 0, s23 <= s21 + s22 + 1, s24 >= 0, s24 <= s20 + s23 + 1, x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 24 + 4*s25 + 4*s27 + 21*x_115 + 8*x_115^2 + 21*x_215 + 8*x_215^2 }-> s28 :|: s25 >= 0, s25 <= x_115 + 1, s26 >= 0, s26 <= x_215 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 0 + 1, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 24 + 4*s29 + 21*x_121 + 8*x_121^2 + 21*x_221 + 8*x_221^2 }-> s32 :|: s29 >= 0, s29 <= x_121 + 1, s30 >= 0, s30 <= x_221 + 1, s31 >= 0, s31 <= s29 + s30 + 1, s32 >= 0, s32 <= 0 + s31 + 1, z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 37 + 4*s135 + 4*s138 + 21*x_123 + 8*x_123^2 + 21*x_125 + 8*x_125^2 + 21*x_223 + 8*x_223^2 + 21*x_225 + 8*x_225^2 }-> s141 :|: s135 >= 0, s135 <= x_123 + 1, s136 >= 0, s136 <= x_223 + 1, s137 >= 0, s137 <= s135 + s136 + 1, s138 >= 0, s138 <= x_125 + 1, s139 >= 0, s139 <= x_225 + 1, s140 >= 0, s140 <= s138 + s139 + 1, s141 >= 0, s141 <= s137 + s140 + 1, x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s142 + 21*x_123 + 8*x_123^2 + 21*x_126 + 8*x_126^2 + 21*x_223 + 8*x_223^2 + 21*x_226 + 8*x_226^2 }-> s148 :|: s142 >= 0, s142 <= x_123 + 1, s143 >= 0, s143 <= x_223 + 1, s144 >= 0, s144 <= s142 + s143 + 1, s145 >= 0, s145 <= x_126 + 1, s146 >= 0, s146 <= x_226 + 1, s147 >= 0, s147 <= s145 + s146 + 1, s148 >= 0, s148 <= s144 + s147 + 1, x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 19 + 4*s149 + 21*x_123 + 8*x_123^2 + 21*x_223 + 8*x_223^2 }-> s152 :|: s149 >= 0, s149 <= x_123 + 1, s150 >= 0, s150 <= x_223 + 1, s151 >= 0, s151 <= s149 + s150 + 1, s152 >= 0, s152 <= s151 + 0 + 1, x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s156 + 21*x_124 + 8*x_124^2 + 21*x_127 + 8*x_127^2 + 21*x_224 + 8*x_224^2 + 21*x_227 + 8*x_227^2 }-> s159 :|: s153 >= 0, s153 <= x_124 + 1, s154 >= 0, s154 <= x_224 + 1, s155 >= 0, s155 <= s153 + s154 + 1, s156 >= 0, s156 <= x_127 + 1, s157 >= 0, s157 <= x_227 + 1, s158 >= 0, s158 <= s156 + s157 + 1, s159 >= 0, s159 <= s155 + s158 + 1, x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 27 + 21*x_124 + 8*x_124^2 + 21*x_128 + 8*x_128^2 + 21*x_224 + 8*x_224^2 + 21*x_228 + 8*x_228^2 }-> s166 :|: s160 >= 0, s160 <= x_124 + 1, s161 >= 0, s161 <= x_224 + 1, s162 >= 0, s162 <= s160 + s161 + 1, s163 >= 0, s163 <= x_128 + 1, s164 >= 0, s164 <= x_228 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s162 + s165 + 1, x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 14 + 21*x_124 + 8*x_124^2 + 21*x_224 + 8*x_224^2 }-> s170 :|: s167 >= 0, s167 <= x_124 + 1, s168 >= 0, s168 <= x_224 + 1, s169 >= 0, s169 <= s167 + s168 + 1, s170 >= 0, s170 <= s169 + 0 + 1, x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 19 + 4*s171 + 21*x_129 + 8*x_129^2 + 21*x_229 + 8*x_229^2 }-> s174 :|: s171 >= 0, s171 <= x_129 + 1, s172 >= 0, s172 <= x_229 + 1, s173 >= 0, s173 <= s171 + s172 + 1, s174 >= 0, s174 <= 0 + s173 + 1, z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 14 + 21*x_130 + 8*x_130^2 + 21*x_230 + 8*x_230^2 }-> s178 :|: s175 >= 0, s175 <= x_130 + 1, s176 >= 0, s176 <= x_230 + 1, s177 >= 0, s177 <= s175 + s176 + 1, s178 >= 0, s178 <= 0 + s177 + 1, z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [6 + 21*z' + 8*z'^2], size: O(n^1) [1 + z']
  encode_-: runtime: O(n^2) [218 + 264*z' + 96*z'^2 + 264*z'' + 96*z''^2], size: O(n^1) [5 + z' + z'']

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(49) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(50)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 19 + 4*s98 + 21*x_113 + 8*x_113^2 + 21*x_213 + 8*x_213^2 }-> s101 :|: s98 >= 0, s98 <= x_113 + 1, s99 >= 0, s99 <= x_213 + 1, s100 >= 0, s100 <= s98 + s99 + 1, s101 >= 0, s101 <= 0 + s100 + 1, x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 14 + 21*x_114 + 8*x_114^2 + 21*x_214 + 8*x_214^2 }-> s105 :|: s102 >= 0, s102 <= x_114 + 1, s103 >= 0, s103 <= x_214 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= 0 + s104 + 1, x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 24 + 4*s10 + 4*s12 + 21*x_1' + 8*x_1'^2 + 21*x_2' + 8*x_2'^2 }-> s13 :|: s10 >= 0, s10 <= x_1' + 1, s11 >= 0, s11 <= x_2' + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= s12 + 0 + 1, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 24 + 4*s14 + 21*x_15 + 8*x_15^2 + 21*x_25 + 8*x_25^2 }-> s17 :|: s14 >= 0, s14 <= x_15 + 1, s15 >= 0, s15 <= x_25 + 1, s16 >= 0, s16 <= s14 + s15 + 1, s17 >= 0, s17 <= 0 + s16 + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 37 + 4*s33 + 4*s35 + 21*x_1' + 8*x_1'^2 + 21*x_12 + 8*x_12^2 + 21*x_2' + 8*x_2'^2 + 21*x_22 + 8*x_22^2 }-> s39 :|: s33 >= 0, s33 <= x_1' + 1, s34 >= 0, s34 <= x_2' + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= x_12 + 1, s37 >= 0, s37 <= x_22 + 1, s38 >= 0, s38 <= s36 + s37 + 1, s39 >= 0, s39 <= s35 + s38 + 1, x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 37 + 4*s42 + 4*s43 + 21*x_1'' + 8*x_1''^2 + 21*x_13 + 8*x_13^2 + 21*x_2'' + 8*x_2''^2 + 21*x_23 + 8*x_23^2 }-> s46 :|: s40 >= 0, s40 <= x_1'' + 1, s41 >= 0, s41 <= x_2'' + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= x_13 + 1, s44 >= 0, s44 <= x_23 + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s42 + s45 + 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') -{ 32 + 4*s49 + 21*x_1'' + 8*x_1''^2 + 21*x_14 + 8*x_14^2 + 21*x_2'' + 8*x_2''^2 + 21*x_24 + 8*x_24^2 }-> s53 :|: s47 >= 0, s47 <= x_1'' + 1, s48 >= 0, s48 <= x_2'' + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= x_14 + 1, s51 >= 0, s51 <= x_24 + 1, s52 >= 0, s52 <= s50 + s51 + 1, s53 >= 0, s53 <= s49 + s52 + 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') -{ 19 + 4*s56 + 21*x_1'' + 8*x_1''^2 + 21*x_2'' + 8*x_2''^2 }-> s57 :|: s54 >= 0, s54 <= x_1'' + 1, s55 >= 0, s55 <= x_2'' + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 0 + 1, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 19 + 21*x_16 + 8*x_16^2 + 21*x_26 + 8*x_26^2 }-> s61 :|: s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= x_26 + 1, s60 >= 0, s60 <= s58 + s59 + 1, s61 >= 0, s61 <= 0 + s60 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 37 + 4*s62 + 4*s65 + 21*x_17 + 8*x_17^2 + 21*x_19 + 8*x_19^2 + 21*x_27 + 8*x_27^2 + 21*x_29 + 8*x_29^2 }-> s68 :|: s62 >= 0, s62 <= x_17 + 1, s63 >= 0, s63 <= x_27 + 1, s64 >= 0, s64 <= s62 + s63 + 1, s65 >= 0, s65 <= x_19 + 1, s66 >= 0, s66 <= x_29 + 1, s67 >= 0, s67 <= s65 + s66 + 1, s68 >= 0, s68 <= s64 + s67 + 1, z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 32 + 4*s69 + 21*x_110 + 8*x_110^2 + 21*x_17 + 8*x_17^2 + 21*x_210 + 8*x_210^2 + 21*x_27 + 8*x_27^2 }-> s75 :|: s69 >= 0, s69 <= x_17 + 1, s70 >= 0, s70 <= x_27 + 1, s71 >= 0, s71 <= s69 + s70 + 1, s72 >= 0, s72 <= x_110 + 1, s73 >= 0, s73 <= x_210 + 1, s74 >= 0, s74 <= s72 + s73 + 1, s75 >= 0, s75 <= s71 + s74 + 1, x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 19 + 4*s76 + 21*x_17 + 8*x_17^2 + 21*x_27 + 8*x_27^2 }-> s79 :|: s76 >= 0, s76 <= x_17 + 1, s77 >= 0, s77 <= x_27 + 1, s78 >= 0, s78 <= s76 + s77 + 1, s79 >= 0, s79 <= s78 + 0 + 1, x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 32 + 4*s83 + 21*x_111 + 8*x_111^2 + 21*x_18 + 8*x_18^2 + 21*x_211 + 8*x_211^2 + 21*x_28 + 8*x_28^2 }-> s86 :|: s80 >= 0, s80 <= x_18 + 1, s81 >= 0, s81 <= x_28 + 1, s82 >= 0, s82 <= s80 + s81 + 1, s83 >= 0, s83 <= x_111 + 1, s84 >= 0, s84 <= x_211 + 1, s85 >= 0, s85 <= s83 + s84 + 1, s86 >= 0, s86 <= s82 + s85 + 1, z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 42 + 4*s3 + 4*s5 + 4*s6 + 21*x_1' + 8*x_1'^2 + 21*x_11 + 8*x_11^2 + 21*x_2' + 8*x_2'^2 + 21*x_21 + 8*x_21^2 }-> s9 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= s3 + s4 + 1, s6 >= 0, s6 <= x_11 + 1, s7 >= 0, s7 <= x_21 + 1, s8 >= 0, s8 <= s6 + s7 + 1, s9 >= 0, s9 <= s5 + s8 + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 27 + 21*x_112 + 8*x_112^2 + 21*x_18 + 8*x_18^2 + 21*x_212 + 8*x_212^2 + 21*x_28 + 8*x_28^2 }-> s93 :|: s87 >= 0, s87 <= x_18 + 1, s88 >= 0, s88 <= x_28 + 1, s89 >= 0, s89 <= s87 + s88 + 1, s90 >= 0, s90 <= x_112 + 1, s91 >= 0, s91 <= x_212 + 1, s92 >= 0, s92 <= s90 + s91 + 1, s93 >= 0, s93 <= s89 + s92 + 1, x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 14 + 21*x_18 + 8*x_18^2 + 21*x_28 + 8*x_28^2 }-> s97 :|: s94 >= 0, s94 <= x_18 + 1, s95 >= 0, s95 <= x_28 + 1, s96 >= 0, s96 <= s94 + s95 + 1, s97 >= 0, s97 <= s96 + 0 + 1, x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 37 + 4*s106 + 4*s108 + 21*x_115 + 8*x_115^2 + 21*x_118 + 8*x_118^2 + 21*x_215 + 8*x_215^2 + 21*x_218 + 8*x_218^2 }-> s112 :|: s106 >= 0, s106 <= x_115 + 1, s107 >= 0, s107 <= x_215 + 1, s108 >= 0, s108 <= s106 + s107 + 1, s109 >= 0, s109 <= x_118 + 1, s110 >= 0, s110 <= x_218 + 1, s111 >= 0, s111 <= s109 + s110 + 1, s112 >= 0, s112 <= s108 + s111 + 1, z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 37 + 4*s115 + 4*s116 + 21*x_116 + 8*x_116^2 + 21*x_119 + 8*x_119^2 + 21*x_216 + 8*x_216^2 + 21*x_219 + 8*x_219^2 }-> s119 :|: s113 >= 0, s113 <= x_116 + 1, s114 >= 0, s114 <= x_216 + 1, s115 >= 0, s115 <= s113 + s114 + 1, s116 >= 0, s116 <= x_119 + 1, s117 >= 0, s117 <= x_219 + 1, s118 >= 0, s118 <= s116 + s117 + 1, s119 >= 0, s119 <= s115 + s118 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 32 + 4*s122 + 21*x_116 + 8*x_116^2 + 21*x_120 + 8*x_120^2 + 21*x_216 + 8*x_216^2 + 21*x_220 + 8*x_220^2 }-> s126 :|: s120 >= 0, s120 <= x_116 + 1, s121 >= 0, s121 <= x_216 + 1, s122 >= 0, s122 <= s120 + s121 + 1, s123 >= 0, s123 <= x_120 + 1, s124 >= 0, s124 <= x_220 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= s122 + s125 + 1, x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 19 + 4*s129 + 21*x_116 + 8*x_116^2 + 21*x_216 + 8*x_216^2 }-> s130 :|: s127 >= 0, s127 <= x_116 + 1, s128 >= 0, s128 <= x_216 + 1, s129 >= 0, s129 <= s127 + s128 + 1, s130 >= 0, s130 <= s129 + 0 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 19 + 21*x_122 + 8*x_122^2 + 21*x_222 + 8*x_222^2 }-> s134 :|: s131 >= 0, s131 <= x_122 + 1, s132 >= 0, s132 <= x_222 + 1, s133 >= 0, s133 <= s131 + s132 + 1, s134 >= 0, s134 <= 0 + s133 + 1, z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 42 + 4*s18 + 4*s20 + 4*s21 + 21*x_115 + 8*x_115^2 + 21*x_117 + 8*x_117^2 + 21*x_215 + 8*x_215^2 + 21*x_217 + 8*x_217^2 }-> s24 :|: s18 >= 0, s18 <= x_115 + 1, s19 >= 0, s19 <= x_215 + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= x_117 + 1, s22 >= 0, s22 <= x_217 + 1, s23 >= 0, s23 <= s21 + s22 + 1, s24 >= 0, s24 <= s20 + s23 + 1, x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 24 + 4*s25 + 4*s27 + 21*x_115 + 8*x_115^2 + 21*x_215 + 8*x_215^2 }-> s28 :|: s25 >= 0, s25 <= x_115 + 1, s26 >= 0, s26 <= x_215 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 0 + 1, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 24 + 4*s29 + 21*x_121 + 8*x_121^2 + 21*x_221 + 8*x_221^2 }-> s32 :|: s29 >= 0, s29 <= x_121 + 1, s30 >= 0, s30 <= x_221 + 1, s31 >= 0, s31 <= s29 + s30 + 1, s32 >= 0, s32 <= 0 + s31 + 1, z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 37 + 4*s135 + 4*s138 + 21*x_123 + 8*x_123^2 + 21*x_125 + 8*x_125^2 + 21*x_223 + 8*x_223^2 + 21*x_225 + 8*x_225^2 }-> s141 :|: s135 >= 0, s135 <= x_123 + 1, s136 >= 0, s136 <= x_223 + 1, s137 >= 0, s137 <= s135 + s136 + 1, s138 >= 0, s138 <= x_125 + 1, s139 >= 0, s139 <= x_225 + 1, s140 >= 0, s140 <= s138 + s139 + 1, s141 >= 0, s141 <= s137 + s140 + 1, x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s142 + 21*x_123 + 8*x_123^2 + 21*x_126 + 8*x_126^2 + 21*x_223 + 8*x_223^2 + 21*x_226 + 8*x_226^2 }-> s148 :|: s142 >= 0, s142 <= x_123 + 1, s143 >= 0, s143 <= x_223 + 1, s144 >= 0, s144 <= s142 + s143 + 1, s145 >= 0, s145 <= x_126 + 1, s146 >= 0, s146 <= x_226 + 1, s147 >= 0, s147 <= s145 + s146 + 1, s148 >= 0, s148 <= s144 + s147 + 1, x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 19 + 4*s149 + 21*x_123 + 8*x_123^2 + 21*x_223 + 8*x_223^2 }-> s152 :|: s149 >= 0, s149 <= x_123 + 1, s150 >= 0, s150 <= x_223 + 1, s151 >= 0, s151 <= s149 + s150 + 1, s152 >= 0, s152 <= s151 + 0 + 1, x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s156 + 21*x_124 + 8*x_124^2 + 21*x_127 + 8*x_127^2 + 21*x_224 + 8*x_224^2 + 21*x_227 + 8*x_227^2 }-> s159 :|: s153 >= 0, s153 <= x_124 + 1, s154 >= 0, s154 <= x_224 + 1, s155 >= 0, s155 <= s153 + s154 + 1, s156 >= 0, s156 <= x_127 + 1, s157 >= 0, s157 <= x_227 + 1, s158 >= 0, s158 <= s156 + s157 + 1, s159 >= 0, s159 <= s155 + s158 + 1, x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 27 + 21*x_124 + 8*x_124^2 + 21*x_128 + 8*x_128^2 + 21*x_224 + 8*x_224^2 + 21*x_228 + 8*x_228^2 }-> s166 :|: s160 >= 0, s160 <= x_124 + 1, s161 >= 0, s161 <= x_224 + 1, s162 >= 0, s162 <= s160 + s161 + 1, s163 >= 0, s163 <= x_128 + 1, s164 >= 0, s164 <= x_228 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s162 + s165 + 1, x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 14 + 21*x_124 + 8*x_124^2 + 21*x_224 + 8*x_224^2 }-> s170 :|: s167 >= 0, s167 <= x_124 + 1, s168 >= 0, s168 <= x_224 + 1, s169 >= 0, s169 <= s167 + s168 + 1, s170 >= 0, s170 <= s169 + 0 + 1, x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 19 + 4*s171 + 21*x_129 + 8*x_129^2 + 21*x_229 + 8*x_229^2 }-> s174 :|: s171 >= 0, s171 <= x_129 + 1, s172 >= 0, s172 <= x_229 + 1, s173 >= 0, s173 <= s171 + s172 + 1, s174 >= 0, s174 <= 0 + s173 + 1, z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 14 + 21*x_130 + 8*x_130^2 + 21*x_230 + 8*x_230^2 }-> s178 :|: s175 >= 0, s175 <= x_130 + 1, s176 >= 0, s176 <= x_230 + 1, s177 >= 0, s177 <= s175 + s176 + 1, s178 >= 0, s178 <= 0 + s177 + 1, z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [6 + 21*z' + 8*z'^2], size: O(n^1) [1 + z']
  encode_-: runtime: O(n^2) [218 + 264*z' + 96*z'^2 + 264*z'' + 96*z''^2], size: O(n^1) [5 + z' + z'']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 19 + 4*s98 + 21*x_113 + 8*x_113^2 + 21*x_213 + 8*x_213^2 }-> s101 :|: s98 >= 0, s98 <= x_113 + 1, s99 >= 0, s99 <= x_213 + 1, s100 >= 0, s100 <= s98 + s99 + 1, s101 >= 0, s101 <= 0 + s100 + 1, x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 14 + 21*x_114 + 8*x_114^2 + 21*x_214 + 8*x_214^2 }-> s105 :|: s102 >= 0, s102 <= x_114 + 1, s103 >= 0, s103 <= x_214 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= 0 + s104 + 1, x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 24 + 4*s10 + 4*s12 + 21*x_1' + 8*x_1'^2 + 21*x_2' + 8*x_2'^2 }-> s13 :|: s10 >= 0, s10 <= x_1' + 1, s11 >= 0, s11 <= x_2' + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= s12 + 0 + 1, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 24 + 4*s14 + 21*x_15 + 8*x_15^2 + 21*x_25 + 8*x_25^2 }-> s17 :|: s14 >= 0, s14 <= x_15 + 1, s15 >= 0, s15 <= x_25 + 1, s16 >= 0, s16 <= s14 + s15 + 1, s17 >= 0, s17 <= 0 + s16 + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 37 + 4*s33 + 4*s35 + 21*x_1' + 8*x_1'^2 + 21*x_12 + 8*x_12^2 + 21*x_2' + 8*x_2'^2 + 21*x_22 + 8*x_22^2 }-> s39 :|: s33 >= 0, s33 <= x_1' + 1, s34 >= 0, s34 <= x_2' + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= x_12 + 1, s37 >= 0, s37 <= x_22 + 1, s38 >= 0, s38 <= s36 + s37 + 1, s39 >= 0, s39 <= s35 + s38 + 1, x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 37 + 4*s42 + 4*s43 + 21*x_1'' + 8*x_1''^2 + 21*x_13 + 8*x_13^2 + 21*x_2'' + 8*x_2''^2 + 21*x_23 + 8*x_23^2 }-> s46 :|: s40 >= 0, s40 <= x_1'' + 1, s41 >= 0, s41 <= x_2'' + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= x_13 + 1, s44 >= 0, s44 <= x_23 + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s42 + s45 + 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') -{ 32 + 4*s49 + 21*x_1'' + 8*x_1''^2 + 21*x_14 + 8*x_14^2 + 21*x_2'' + 8*x_2''^2 + 21*x_24 + 8*x_24^2 }-> s53 :|: s47 >= 0, s47 <= x_1'' + 1, s48 >= 0, s48 <= x_2'' + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= x_14 + 1, s51 >= 0, s51 <= x_24 + 1, s52 >= 0, s52 <= s50 + s51 + 1, s53 >= 0, s53 <= s49 + s52 + 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') -{ 19 + 4*s56 + 21*x_1'' + 8*x_1''^2 + 21*x_2'' + 8*x_2''^2 }-> s57 :|: s54 >= 0, s54 <= x_1'' + 1, s55 >= 0, s55 <= x_2'' + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 0 + 1, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 19 + 21*x_16 + 8*x_16^2 + 21*x_26 + 8*x_26^2 }-> s61 :|: s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= x_26 + 1, s60 >= 0, s60 <= s58 + s59 + 1, s61 >= 0, s61 <= 0 + s60 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 37 + 4*s62 + 4*s65 + 21*x_17 + 8*x_17^2 + 21*x_19 + 8*x_19^2 + 21*x_27 + 8*x_27^2 + 21*x_29 + 8*x_29^2 }-> s68 :|: s62 >= 0, s62 <= x_17 + 1, s63 >= 0, s63 <= x_27 + 1, s64 >= 0, s64 <= s62 + s63 + 1, s65 >= 0, s65 <= x_19 + 1, s66 >= 0, s66 <= x_29 + 1, s67 >= 0, s67 <= s65 + s66 + 1, s68 >= 0, s68 <= s64 + s67 + 1, z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 32 + 4*s69 + 21*x_110 + 8*x_110^2 + 21*x_17 + 8*x_17^2 + 21*x_210 + 8*x_210^2 + 21*x_27 + 8*x_27^2 }-> s75 :|: s69 >= 0, s69 <= x_17 + 1, s70 >= 0, s70 <= x_27 + 1, s71 >= 0, s71 <= s69 + s70 + 1, s72 >= 0, s72 <= x_110 + 1, s73 >= 0, s73 <= x_210 + 1, s74 >= 0, s74 <= s72 + s73 + 1, s75 >= 0, s75 <= s71 + s74 + 1, x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 19 + 4*s76 + 21*x_17 + 8*x_17^2 + 21*x_27 + 8*x_27^2 }-> s79 :|: s76 >= 0, s76 <= x_17 + 1, s77 >= 0, s77 <= x_27 + 1, s78 >= 0, s78 <= s76 + s77 + 1, s79 >= 0, s79 <= s78 + 0 + 1, x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 32 + 4*s83 + 21*x_111 + 8*x_111^2 + 21*x_18 + 8*x_18^2 + 21*x_211 + 8*x_211^2 + 21*x_28 + 8*x_28^2 }-> s86 :|: s80 >= 0, s80 <= x_18 + 1, s81 >= 0, s81 <= x_28 + 1, s82 >= 0, s82 <= s80 + s81 + 1, s83 >= 0, s83 <= x_111 + 1, s84 >= 0, s84 <= x_211 + 1, s85 >= 0, s85 <= s83 + s84 + 1, s86 >= 0, s86 <= s82 + s85 + 1, z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 42 + 4*s3 + 4*s5 + 4*s6 + 21*x_1' + 8*x_1'^2 + 21*x_11 + 8*x_11^2 + 21*x_2' + 8*x_2'^2 + 21*x_21 + 8*x_21^2 }-> s9 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= s3 + s4 + 1, s6 >= 0, s6 <= x_11 + 1, s7 >= 0, s7 <= x_21 + 1, s8 >= 0, s8 <= s6 + s7 + 1, s9 >= 0, s9 <= s5 + s8 + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 27 + 21*x_112 + 8*x_112^2 + 21*x_18 + 8*x_18^2 + 21*x_212 + 8*x_212^2 + 21*x_28 + 8*x_28^2 }-> s93 :|: s87 >= 0, s87 <= x_18 + 1, s88 >= 0, s88 <= x_28 + 1, s89 >= 0, s89 <= s87 + s88 + 1, s90 >= 0, s90 <= x_112 + 1, s91 >= 0, s91 <= x_212 + 1, s92 >= 0, s92 <= s90 + s91 + 1, s93 >= 0, s93 <= s89 + s92 + 1, x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 14 + 21*x_18 + 8*x_18^2 + 21*x_28 + 8*x_28^2 }-> s97 :|: s94 >= 0, s94 <= x_18 + 1, s95 >= 0, s95 <= x_28 + 1, s96 >= 0, s96 <= s94 + s95 + 1, s97 >= 0, s97 <= s96 + 0 + 1, x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 37 + 4*s106 + 4*s108 + 21*x_115 + 8*x_115^2 + 21*x_118 + 8*x_118^2 + 21*x_215 + 8*x_215^2 + 21*x_218 + 8*x_218^2 }-> s112 :|: s106 >= 0, s106 <= x_115 + 1, s107 >= 0, s107 <= x_215 + 1, s108 >= 0, s108 <= s106 + s107 + 1, s109 >= 0, s109 <= x_118 + 1, s110 >= 0, s110 <= x_218 + 1, s111 >= 0, s111 <= s109 + s110 + 1, s112 >= 0, s112 <= s108 + s111 + 1, z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 37 + 4*s115 + 4*s116 + 21*x_116 + 8*x_116^2 + 21*x_119 + 8*x_119^2 + 21*x_216 + 8*x_216^2 + 21*x_219 + 8*x_219^2 }-> s119 :|: s113 >= 0, s113 <= x_116 + 1, s114 >= 0, s114 <= x_216 + 1, s115 >= 0, s115 <= s113 + s114 + 1, s116 >= 0, s116 <= x_119 + 1, s117 >= 0, s117 <= x_219 + 1, s118 >= 0, s118 <= s116 + s117 + 1, s119 >= 0, s119 <= s115 + s118 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 32 + 4*s122 + 21*x_116 + 8*x_116^2 + 21*x_120 + 8*x_120^2 + 21*x_216 + 8*x_216^2 + 21*x_220 + 8*x_220^2 }-> s126 :|: s120 >= 0, s120 <= x_116 + 1, s121 >= 0, s121 <= x_216 + 1, s122 >= 0, s122 <= s120 + s121 + 1, s123 >= 0, s123 <= x_120 + 1, s124 >= 0, s124 <= x_220 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= s122 + s125 + 1, x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 19 + 4*s129 + 21*x_116 + 8*x_116^2 + 21*x_216 + 8*x_216^2 }-> s130 :|: s127 >= 0, s127 <= x_116 + 1, s128 >= 0, s128 <= x_216 + 1, s129 >= 0, s129 <= s127 + s128 + 1, s130 >= 0, s130 <= s129 + 0 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 19 + 21*x_122 + 8*x_122^2 + 21*x_222 + 8*x_222^2 }-> s134 :|: s131 >= 0, s131 <= x_122 + 1, s132 >= 0, s132 <= x_222 + 1, s133 >= 0, s133 <= s131 + s132 + 1, s134 >= 0, s134 <= 0 + s133 + 1, z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 42 + 4*s18 + 4*s20 + 4*s21 + 21*x_115 + 8*x_115^2 + 21*x_117 + 8*x_117^2 + 21*x_215 + 8*x_215^2 + 21*x_217 + 8*x_217^2 }-> s24 :|: s18 >= 0, s18 <= x_115 + 1, s19 >= 0, s19 <= x_215 + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= x_117 + 1, s22 >= 0, s22 <= x_217 + 1, s23 >= 0, s23 <= s21 + s22 + 1, s24 >= 0, s24 <= s20 + s23 + 1, x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 24 + 4*s25 + 4*s27 + 21*x_115 + 8*x_115^2 + 21*x_215 + 8*x_215^2 }-> s28 :|: s25 >= 0, s25 <= x_115 + 1, s26 >= 0, s26 <= x_215 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 0 + 1, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 24 + 4*s29 + 21*x_121 + 8*x_121^2 + 21*x_221 + 8*x_221^2 }-> s32 :|: s29 >= 0, s29 <= x_121 + 1, s30 >= 0, s30 <= x_221 + 1, s31 >= 0, s31 <= s29 + s30 + 1, s32 >= 0, s32 <= 0 + s31 + 1, z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 37 + 4*s135 + 4*s138 + 21*x_123 + 8*x_123^2 + 21*x_125 + 8*x_125^2 + 21*x_223 + 8*x_223^2 + 21*x_225 + 8*x_225^2 }-> s141 :|: s135 >= 0, s135 <= x_123 + 1, s136 >= 0, s136 <= x_223 + 1, s137 >= 0, s137 <= s135 + s136 + 1, s138 >= 0, s138 <= x_125 + 1, s139 >= 0, s139 <= x_225 + 1, s140 >= 0, s140 <= s138 + s139 + 1, s141 >= 0, s141 <= s137 + s140 + 1, x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s142 + 21*x_123 + 8*x_123^2 + 21*x_126 + 8*x_126^2 + 21*x_223 + 8*x_223^2 + 21*x_226 + 8*x_226^2 }-> s148 :|: s142 >= 0, s142 <= x_123 + 1, s143 >= 0, s143 <= x_223 + 1, s144 >= 0, s144 <= s142 + s143 + 1, s145 >= 0, s145 <= x_126 + 1, s146 >= 0, s146 <= x_226 + 1, s147 >= 0, s147 <= s145 + s146 + 1, s148 >= 0, s148 <= s144 + s147 + 1, x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 19 + 4*s149 + 21*x_123 + 8*x_123^2 + 21*x_223 + 8*x_223^2 }-> s152 :|: s149 >= 0, s149 <= x_123 + 1, s150 >= 0, s150 <= x_223 + 1, s151 >= 0, s151 <= s149 + s150 + 1, s152 >= 0, s152 <= s151 + 0 + 1, x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s156 + 21*x_124 + 8*x_124^2 + 21*x_127 + 8*x_127^2 + 21*x_224 + 8*x_224^2 + 21*x_227 + 8*x_227^2 }-> s159 :|: s153 >= 0, s153 <= x_124 + 1, s154 >= 0, s154 <= x_224 + 1, s155 >= 0, s155 <= s153 + s154 + 1, s156 >= 0, s156 <= x_127 + 1, s157 >= 0, s157 <= x_227 + 1, s158 >= 0, s158 <= s156 + s157 + 1, s159 >= 0, s159 <= s155 + s158 + 1, x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 27 + 21*x_124 + 8*x_124^2 + 21*x_128 + 8*x_128^2 + 21*x_224 + 8*x_224^2 + 21*x_228 + 8*x_228^2 }-> s166 :|: s160 >= 0, s160 <= x_124 + 1, s161 >= 0, s161 <= x_224 + 1, s162 >= 0, s162 <= s160 + s161 + 1, s163 >= 0, s163 <= x_128 + 1, s164 >= 0, s164 <= x_228 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s162 + s165 + 1, x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 14 + 21*x_124 + 8*x_124^2 + 21*x_224 + 8*x_224^2 }-> s170 :|: s167 >= 0, s167 <= x_124 + 1, s168 >= 0, s168 <= x_224 + 1, s169 >= 0, s169 <= s167 + s168 + 1, s170 >= 0, s170 <= s169 + 0 + 1, x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 19 + 4*s171 + 21*x_129 + 8*x_129^2 + 21*x_229 + 8*x_229^2 }-> s174 :|: s171 >= 0, s171 <= x_129 + 1, s172 >= 0, s172 <= x_229 + 1, s173 >= 0, s173 <= s171 + s172 + 1, s174 >= 0, s174 <= 0 + s173 + 1, z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 14 + 21*x_130 + 8*x_130^2 + 21*x_230 + 8*x_230^2 }-> s178 :|: s175 >= 0, s175 <= x_130 + 1, s176 >= 0, s176 <= x_230 + 1, s177 >= 0, s177 <= s175 + s176 + 1, s178 >= 0, s178 <= 0 + s177 + 1, z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {encode_+}
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [6 + 21*z' + 8*z'^2], size: O(n^1) [1 + z']
  encode_-: runtime: O(n^2) [218 + 264*z' + 96*z'^2 + 264*z'' + 96*z''^2], size: O(n^1) [5 + z' + z'']
  encode_+: runtime: ?, size: O(n^1) [5 + z' + z'']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_+
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 288 + 312*z' + 96*z'^2 + 264*z'' + 96*z''^2

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z') -{ 6 }-> s :|: s >= 0, s <= 0 + 0 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 19 + 4*s98 + 21*x_113 + 8*x_113^2 + 21*x_213 + 8*x_213^2 }-> s101 :|: s98 >= 0, s98 <= x_113 + 1, s99 >= 0, s99 <= x_213 + 1, s100 >= 0, s100 <= s98 + s99 + 1, s101 >= 0, s101 <= 0 + s100 + 1, x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = 1 + x_1 + (1 + x_113 + x_213)
   encArg(z') -{ 14 + 21*x_114 + 8*x_114^2 + 21*x_214 + 8*x_214^2 }-> s105 :|: s102 >= 0, s102 <= x_114 + 1, s103 >= 0, s103 <= x_214 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= 0 + s104 + 1, x_1 >= 0, x_114 >= 0, x_214 >= 0, z' = 1 + x_1 + (1 + x_114 + x_214)
   encArg(z') -{ 24 + 4*s10 + 4*s12 + 21*x_1' + 8*x_1'^2 + 21*x_2' + 8*x_2'^2 }-> s13 :|: s10 >= 0, s10 <= x_1' + 1, s11 >= 0, s11 <= x_2' + 1, s12 >= 0, s12 <= s10 + s11 + 1, s13 >= 0, s13 <= s12 + 0 + 1, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 24 + 4*s14 + 21*x_15 + 8*x_15^2 + 21*x_25 + 8*x_25^2 }-> s17 :|: s14 >= 0, s14 <= x_15 + 1, s15 >= 0, s15 <= x_25 + 1, s16 >= 0, s16 <= s14 + s15 + 1, s17 >= 0, s17 <= 0 + s16 + 1, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 37 + 4*s33 + 4*s35 + 21*x_1' + 8*x_1'^2 + 21*x_12 + 8*x_12^2 + 21*x_2' + 8*x_2'^2 + 21*x_22 + 8*x_22^2 }-> s39 :|: s33 >= 0, s33 <= x_1' + 1, s34 >= 0, s34 <= x_2' + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= x_12 + 1, s37 >= 0, s37 <= x_22 + 1, s38 >= 0, s38 <= s36 + s37 + 1, s39 >= 0, s39 <= s35 + s38 + 1, x_2' >= 0, x_1' >= 0, x_12 >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_12 + x_22), x_22 >= 0
   encArg(z') -{ 37 + 4*s42 + 4*s43 + 21*x_1'' + 8*x_1''^2 + 21*x_13 + 8*x_13^2 + 21*x_2'' + 8*x_2''^2 + 21*x_23 + 8*x_23^2 }-> s46 :|: s40 >= 0, s40 <= x_1'' + 1, s41 >= 0, s41 <= x_2'' + 1, s42 >= 0, s42 <= s40 + s41 + 1, s43 >= 0, s43 <= x_13 + 1, s44 >= 0, s44 <= x_23 + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s42 + s45 + 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') -{ 32 + 4*s49 + 21*x_1'' + 8*x_1''^2 + 21*x_14 + 8*x_14^2 + 21*x_2'' + 8*x_2''^2 + 21*x_24 + 8*x_24^2 }-> s53 :|: s47 >= 0, s47 <= x_1'' + 1, s48 >= 0, s48 <= x_2'' + 1, s49 >= 0, s49 <= s47 + s48 + 1, s50 >= 0, s50 <= x_14 + 1, s51 >= 0, s51 <= x_24 + 1, s52 >= 0, s52 <= s50 + s51 + 1, s53 >= 0, s53 <= s49 + s52 + 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') -{ 19 + 4*s56 + 21*x_1'' + 8*x_1''^2 + 21*x_2'' + 8*x_2''^2 }-> s57 :|: s54 >= 0, s54 <= x_1'' + 1, s55 >= 0, s55 <= x_2'' + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 0 + 1, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 19 + 21*x_16 + 8*x_16^2 + 21*x_26 + 8*x_26^2 }-> s61 :|: s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= x_26 + 1, s60 >= 0, s60 <= s58 + s59 + 1, s61 >= 0, s61 <= 0 + s60 + 1, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 37 + 4*s62 + 4*s65 + 21*x_17 + 8*x_17^2 + 21*x_19 + 8*x_19^2 + 21*x_27 + 8*x_27^2 + 21*x_29 + 8*x_29^2 }-> s68 :|: s62 >= 0, s62 <= x_17 + 1, s63 >= 0, s63 <= x_27 + 1, s64 >= 0, s64 <= s62 + s63 + 1, s65 >= 0, s65 <= x_19 + 1, s66 >= 0, s66 <= x_29 + 1, s67 >= 0, s67 <= s65 + s66 + 1, s68 >= 0, s68 <= s64 + s67 + 1, z' = 1 + (1 + x_17 + x_27) + (1 + x_19 + x_29), x_17 >= 0, x_27 >= 0, x_19 >= 0, x_29 >= 0
   encArg(z') -{ 32 + 4*s69 + 21*x_110 + 8*x_110^2 + 21*x_17 + 8*x_17^2 + 21*x_210 + 8*x_210^2 + 21*x_27 + 8*x_27^2 }-> s75 :|: s69 >= 0, s69 <= x_17 + 1, s70 >= 0, s70 <= x_27 + 1, s71 >= 0, s71 <= s69 + s70 + 1, s72 >= 0, s72 <= x_110 + 1, s73 >= 0, s73 <= x_210 + 1, s74 >= 0, s74 <= s72 + s73 + 1, s75 >= 0, s75 <= s71 + s74 + 1, x_17 >= 0, z' = 1 + (1 + x_17 + x_27) + (1 + x_110 + x_210), x_27 >= 0, x_110 >= 0, x_210 >= 0
   encArg(z') -{ 19 + 4*s76 + 21*x_17 + 8*x_17^2 + 21*x_27 + 8*x_27^2 }-> s79 :|: s76 >= 0, s76 <= x_17 + 1, s77 >= 0, s77 <= x_27 + 1, s78 >= 0, s78 <= s76 + s77 + 1, s79 >= 0, s79 <= s78 + 0 + 1, x_17 >= 0, x_27 >= 0, x_2 >= 0, z' = 1 + (1 + x_17 + x_27) + x_2
   encArg(z') -{ 32 + 4*s83 + 21*x_111 + 8*x_111^2 + 21*x_18 + 8*x_18^2 + 21*x_211 + 8*x_211^2 + 21*x_28 + 8*x_28^2 }-> s86 :|: s80 >= 0, s80 <= x_18 + 1, s81 >= 0, s81 <= x_28 + 1, s82 >= 0, s82 <= s80 + s81 + 1, s83 >= 0, s83 <= x_111 + 1, s84 >= 0, s84 <= x_211 + 1, s85 >= 0, s85 <= s83 + s84 + 1, s86 >= 0, s86 <= s82 + s85 + 1, z' = 1 + (1 + x_18 + x_28) + (1 + x_111 + x_211), x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encArg(z') -{ 42 + 4*s3 + 4*s5 + 4*s6 + 21*x_1' + 8*x_1'^2 + 21*x_11 + 8*x_11^2 + 21*x_2' + 8*x_2'^2 + 21*x_21 + 8*x_21^2 }-> s9 :|: s3 >= 0, s3 <= x_1' + 1, s4 >= 0, s4 <= x_2' + 1, s5 >= 0, s5 <= s3 + s4 + 1, s6 >= 0, s6 <= x_11 + 1, s7 >= 0, s7 <= x_21 + 1, s8 >= 0, s8 <= s6 + s7 + 1, s9 >= 0, s9 <= s5 + s8 + 1, x_11 >= 0, x_2' >= 0, x_1' >= 0, z' = 1 + (1 + x_1' + x_2') + (1 + x_11 + x_21), x_21 >= 0
   encArg(z') -{ 27 + 21*x_112 + 8*x_112^2 + 21*x_18 + 8*x_18^2 + 21*x_212 + 8*x_212^2 + 21*x_28 + 8*x_28^2 }-> s93 :|: s87 >= 0, s87 <= x_18 + 1, s88 >= 0, s88 <= x_28 + 1, s89 >= 0, s89 <= s87 + s88 + 1, s90 >= 0, s90 <= x_112 + 1, s91 >= 0, s91 <= x_212 + 1, s92 >= 0, s92 <= s90 + s91 + 1, s93 >= 0, s93 <= s89 + s92 + 1, x_212 >= 0, z' = 1 + (1 + x_18 + x_28) + (1 + x_112 + x_212), x_18 >= 0, x_28 >= 0, x_112 >= 0
   encArg(z') -{ 14 + 21*x_18 + 8*x_18^2 + 21*x_28 + 8*x_28^2 }-> s97 :|: s94 >= 0, s94 <= x_18 + 1, s95 >= 0, s95 <= x_28 + 1, s96 >= 0, s96 <= s94 + s95 + 1, s97 >= 0, s97 <= s96 + 0 + 1, x_2 >= 0, x_18 >= 0, x_28 >= 0, z' = 1 + (1 + x_18 + x_28) + x_2
   encArg(z') -{ 0 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encArg(z') -{ 0 }-> 1 + x0 + x1 :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   encode_+(z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0 + 0 + 1, z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 37 + 4*s106 + 4*s108 + 21*x_115 + 8*x_115^2 + 21*x_118 + 8*x_118^2 + 21*x_215 + 8*x_215^2 + 21*x_218 + 8*x_218^2 }-> s112 :|: s106 >= 0, s106 <= x_115 + 1, s107 >= 0, s107 <= x_215 + 1, s108 >= 0, s108 <= s106 + s107 + 1, s109 >= 0, s109 <= x_118 + 1, s110 >= 0, s110 <= x_218 + 1, s111 >= 0, s111 <= s109 + s110 + 1, s112 >= 0, s112 <= s108 + s111 + 1, z'' = 1 + x_118 + x_218, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_218 >= 0, x_118 >= 0
   encode_+(z', z'') -{ 37 + 4*s115 + 4*s116 + 21*x_116 + 8*x_116^2 + 21*x_119 + 8*x_119^2 + 21*x_216 + 8*x_216^2 + 21*x_219 + 8*x_219^2 }-> s119 :|: s113 >= 0, s113 <= x_116 + 1, s114 >= 0, s114 <= x_216 + 1, s115 >= 0, s115 <= s113 + s114 + 1, s116 >= 0, s116 <= x_119 + 1, s117 >= 0, s117 <= x_219 + 1, s118 >= 0, s118 <= s116 + s117 + 1, s119 >= 0, s119 <= s115 + s118 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' = 1 + x_119 + x_219, x_219 >= 0, x_119 >= 0
   encode_+(z', z'') -{ 32 + 4*s122 + 21*x_116 + 8*x_116^2 + 21*x_120 + 8*x_120^2 + 21*x_216 + 8*x_216^2 + 21*x_220 + 8*x_220^2 }-> s126 :|: s120 >= 0, s120 <= x_116 + 1, s121 >= 0, s121 <= x_216 + 1, s122 >= 0, s122 <= s120 + s121 + 1, s123 >= 0, s123 <= x_120 + 1, s124 >= 0, s124 <= x_220 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= s122 + s125 + 1, x_216 >= 0, x_120 >= 0, z' = 1 + x_116 + x_216, z'' = 1 + x_120 + x_220, x_116 >= 0, x_220 >= 0
   encode_+(z', z'') -{ 19 + 4*s129 + 21*x_116 + 8*x_116^2 + 21*x_216 + 8*x_216^2 }-> s130 :|: s127 >= 0, s127 <= x_116 + 1, s128 >= 0, s128 <= x_216 + 1, s129 >= 0, s129 <= s127 + s128 + 1, s130 >= 0, s130 <= s129 + 0 + 1, x_216 >= 0, z' = 1 + x_116 + x_216, x_116 >= 0, z'' >= 0
   encode_+(z', z'') -{ 19 + 21*x_122 + 8*x_122^2 + 21*x_222 + 8*x_222^2 }-> s134 :|: s131 >= 0, s131 <= x_122 + 1, s132 >= 0, s132 <= x_222 + 1, s133 >= 0, s133 <= s131 + s132 + 1, s134 >= 0, s134 <= 0 + s133 + 1, z' >= 0, x_222 >= 0, x_122 >= 0, z'' = 1 + x_122 + x_222
   encode_+(z', z'') -{ 42 + 4*s18 + 4*s20 + 4*s21 + 21*x_115 + 8*x_115^2 + 21*x_117 + 8*x_117^2 + 21*x_215 + 8*x_215^2 + 21*x_217 + 8*x_217^2 }-> s24 :|: s18 >= 0, s18 <= x_115 + 1, s19 >= 0, s19 <= x_215 + 1, s20 >= 0, s20 <= s18 + s19 + 1, s21 >= 0, s21 <= x_117 + 1, s22 >= 0, s22 <= x_217 + 1, s23 >= 0, s23 <= s21 + s22 + 1, s24 >= 0, s24 <= s20 + s23 + 1, x_117 >= 0, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, x_217 >= 0, z'' = 1 + x_117 + x_217
   encode_+(z', z'') -{ 24 + 4*s25 + 4*s27 + 21*x_115 + 8*x_115^2 + 21*x_215 + 8*x_215^2 }-> s28 :|: s25 >= 0, s25 <= x_115 + 1, s26 >= 0, s26 <= x_215 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 0 + 1, z' = 1 + x_115 + x_215, x_215 >= 0, x_115 >= 0, z'' >= 0
   encode_+(z', z'') -{ 24 + 4*s29 + 21*x_121 + 8*x_121^2 + 21*x_221 + 8*x_221^2 }-> s32 :|: s29 >= 0, s29 <= x_121 + 1, s30 >= 0, s30 <= x_221 + 1, s31 >= 0, s31 <= s29 + s30 + 1, s32 >= 0, s32 <= 0 + s31 + 1, z' >= 0, x_221 >= 0, z'' = 1 + x_121 + x_221, x_121 >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 37 + 4*s135 + 4*s138 + 21*x_123 + 8*x_123^2 + 21*x_125 + 8*x_125^2 + 21*x_223 + 8*x_223^2 + 21*x_225 + 8*x_225^2 }-> s141 :|: s135 >= 0, s135 <= x_123 + 1, s136 >= 0, s136 <= x_223 + 1, s137 >= 0, s137 <= s135 + s136 + 1, s138 >= 0, s138 <= x_125 + 1, s139 >= 0, s139 <= x_225 + 1, s140 >= 0, s140 <= s138 + s139 + 1, s141 >= 0, s141 <= s137 + s140 + 1, x_123 >= 0, x_223 >= 0, x_125 >= 0, z'' = 1 + x_125 + x_225, x_225 >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s142 + 21*x_123 + 8*x_123^2 + 21*x_126 + 8*x_126^2 + 21*x_223 + 8*x_223^2 + 21*x_226 + 8*x_226^2 }-> s148 :|: s142 >= 0, s142 <= x_123 + 1, s143 >= 0, s143 <= x_223 + 1, s144 >= 0, s144 <= s142 + s143 + 1, s145 >= 0, s145 <= x_126 + 1, s146 >= 0, s146 <= x_226 + 1, s147 >= 0, s147 <= s145 + s146 + 1, s148 >= 0, s148 <= s144 + s147 + 1, x_226 >= 0, x_123 >= 0, x_223 >= 0, x_126 >= 0, z' = 1 + x_123 + x_223, z'' = 1 + x_126 + x_226
   encode_-(z', z'') -{ 19 + 4*s149 + 21*x_123 + 8*x_123^2 + 21*x_223 + 8*x_223^2 }-> s152 :|: s149 >= 0, s149 <= x_123 + 1, s150 >= 0, s150 <= x_223 + 1, s151 >= 0, s151 <= s149 + s150 + 1, s152 >= 0, s152 <= s151 + 0 + 1, x_123 >= 0, x_223 >= 0, z'' >= 0, z' = 1 + x_123 + x_223
   encode_-(z', z'') -{ 32 + 4*s156 + 21*x_124 + 8*x_124^2 + 21*x_127 + 8*x_127^2 + 21*x_224 + 8*x_224^2 + 21*x_227 + 8*x_227^2 }-> s159 :|: s153 >= 0, s153 <= x_124 + 1, s154 >= 0, s154 <= x_224 + 1, s155 >= 0, s155 <= s153 + s154 + 1, s156 >= 0, s156 <= x_127 + 1, s157 >= 0, s157 <= x_227 + 1, s158 >= 0, s158 <= s156 + s157 + 1, s159 >= 0, s159 <= s155 + s158 + 1, x_124 >= 0, x_127 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_127 + x_227, x_227 >= 0
   encode_-(z', z'') -{ 27 + 21*x_124 + 8*x_124^2 + 21*x_128 + 8*x_128^2 + 21*x_224 + 8*x_224^2 + 21*x_228 + 8*x_228^2 }-> s166 :|: s160 >= 0, s160 <= x_124 + 1, s161 >= 0, s161 <= x_224 + 1, s162 >= 0, s162 <= s160 + s161 + 1, s163 >= 0, s163 <= x_128 + 1, s164 >= 0, s164 <= x_228 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s162 + s165 + 1, x_124 >= 0, x_128 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' = 1 + x_128 + x_228, x_228 >= 0
   encode_-(z', z'') -{ 14 + 21*x_124 + 8*x_124^2 + 21*x_224 + 8*x_224^2 }-> s170 :|: s167 >= 0, s167 <= x_124 + 1, s168 >= 0, s168 <= x_224 + 1, s169 >= 0, s169 <= s167 + s168 + 1, s170 >= 0, s170 <= s169 + 0 + 1, x_124 >= 0, z' = 1 + x_124 + x_224, x_224 >= 0, z'' >= 0
   encode_-(z', z'') -{ 19 + 4*s171 + 21*x_129 + 8*x_129^2 + 21*x_229 + 8*x_229^2 }-> s174 :|: s171 >= 0, s171 <= x_129 + 1, s172 >= 0, s172 <= x_229 + 1, s173 >= 0, s173 <= s171 + s172 + 1, s174 >= 0, s174 <= 0 + s173 + 1, z' >= 0, z'' = 1 + x_129 + x_229, x_229 >= 0, x_129 >= 0
   encode_-(z', z'') -{ 14 + 21*x_130 + 8*x_130^2 + 21*x_230 + 8*x_230^2 }-> s178 :|: s175 >= 0, s175 <= x_130 + 1, s176 >= 0, s176 <= x_230 + 1, s177 >= 0, s177 <= s175 + s176 + 1, s178 >= 0, s178 <= 0 + s177 + 1, z' >= 0, x_130 >= 0, z'' = 1 + x_130 + x_230, x_230 >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_-(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, v0 >= 0, v1 >= 0, 0 = v1, 0 = v0
   encode_-(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z' >= 0, z'' >= 0, 0 = x1, x0 >= 0, x1 >= 0, 0 = x0
   minus(z', z'') -{ 1 }-> x :|: z' = 1 + x + z'', x >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   minus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 10 + 4*x' }-> s2 :|: s'' >= 0, s'' <= x' + z'' + 1, s1 >= 0, s1 <= s'' + y' + 1, s2 >= 0, s2 <= s1 + y + 1, z'' >= 0, x' >= 0, y >= 0, z' = 1 + (1 + x' + y') + y, y' >= 0
   plus(z', z'') -{ 2 }-> x' :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = y', 1 + x + z'' = 1 + x' + y', x' >= 0, y' >= 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 1 + x + z'' = v0
   plus(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, v0 >= 0, v1 >= 0, y = v1, 0 = v0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 1 + x + z'' = x0
   plus(z', z'') -{ 1 }-> 1 + x0 + x1 :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0, y = x1, x0 >= 0, x1 >= 0, 0 = x0
   plus(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(1) [1], size: O(n^1) [1 + z' + z'']
  plus: runtime: O(n^1) [6 + 4*z'], size: O(n^1) [1 + z' + z'']
  encArg: runtime: O(n^2) [6 + 21*z' + 8*z'^2], size: O(n^1) [1 + z']
  encode_-: runtime: O(n^2) [218 + 264*z' + 96*z'^2 + 264*z'' + 96*z''^2], size: O(n^1) [5 + z' + z'']
  encode_+: runtime: O(n^2) [288 + 312*z' + 96*z'^2 + 264*z'' + 96*z''^2], size: O(n^1) [5 + z' + z'']

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(55) FinalProof (FINISHED)
Computed overall runtime complexity
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(56)
BOUNDS(1, n^2)