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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 178 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 282 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 817 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 1176 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 178 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 688 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 463 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(48) TRS for Loop Detection
    (49) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (50) BEST
        (51) proven lower bound
            (52) LowerBoundPropagationProof [FINISHED, 0 ms]
            (53) BOUNDS(n^1, INF)
        (54) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

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

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(+(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(n^1, n^2).


The TRS R consists of the following rules:

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

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

   encArg(+(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(n^1, n^2).


The TRS R consists of the following rules:

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

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

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

(6)
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:

   *(x, +(y, z)) -> +(*(x, y), *(x, z)) [1]
   encArg(+(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]

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

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

* => times

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

(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:

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_*(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) [0]
   encode_*(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]

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

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

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

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_*(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) [0]
   encode_*(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
times :: +:cons_* -> +:cons_* -> +:cons_*
+ :: +:cons_* -> +:cons_* -> +:cons_*
encArg :: +:cons_* -> +:cons_*
cons_* :: +:cons_* -> +:cons_* -> +:cons_*
encode_* :: +:cons_* -> +:cons_* -> +:cons_*
encode_+ :: +:cons_* -> +:cons_* -> +:cons_*

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

(11) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   times_2
   encArg_1
   encode_*_2
   encode_+_2

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_*(v0, v1) -> const [0]
   encode_+(v0, v1) -> const [0]
   times(v0, v1) -> const [0]

And the following fresh constants:    const

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

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

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

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_*(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) [0]
   encode_*(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> const [0]
   encode_*(v0, v1) -> const [0]
   encode_+(v0, v1) -> const [0]
   times(v0, v1) -> const [0]

The TRS has the following type information:
times :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
+ :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
encArg :: +:cons_*:const -> +:cons_*:const
cons_* :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
encode_* :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
encode_+ :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
const :: +:cons_*:const

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

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

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

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

   times(x, +(y, z)) -> +(times(x, y), times(x, z)) [1]
   encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_*(+(x_1', x_2'), +(x_11, x_21))) -> times(+(encArg(x_1'), encArg(x_2')), +(encArg(x_11), encArg(x_21))) [0]
   encArg(cons_*(+(x_1', x_2'), cons_*(x_12, x_22))) -> times(+(encArg(x_1'), encArg(x_2')), times(encArg(x_12), encArg(x_22))) [0]
   encArg(cons_*(+(x_1', x_2'), x_2)) -> times(+(encArg(x_1'), encArg(x_2')), const) [0]
   encArg(cons_*(cons_*(x_1'', x_2''), +(x_13, x_23))) -> times(times(encArg(x_1''), encArg(x_2'')), +(encArg(x_13), encArg(x_23))) [0]
   encArg(cons_*(cons_*(x_1'', x_2''), cons_*(x_14, x_24))) -> times(times(encArg(x_1''), encArg(x_2'')), times(encArg(x_14), encArg(x_24))) [0]
   encArg(cons_*(cons_*(x_1'', x_2''), x_2)) -> times(times(encArg(x_1''), encArg(x_2'')), const) [0]
   encArg(cons_*(x_1, +(x_15, x_25))) -> times(const, +(encArg(x_15), encArg(x_25))) [0]
   encArg(cons_*(x_1, cons_*(x_16, x_26))) -> times(const, times(encArg(x_16), encArg(x_26))) [0]
   encArg(cons_*(x_1, x_2)) -> times(const, const) [0]
   encode_*(+(x_17, x_27), +(x_19, x_29)) -> times(+(encArg(x_17), encArg(x_27)), +(encArg(x_19), encArg(x_29))) [0]
   encode_*(+(x_17, x_27), cons_*(x_110, x_210)) -> times(+(encArg(x_17), encArg(x_27)), times(encArg(x_110), encArg(x_210))) [0]
   encode_*(+(x_17, x_27), x_2) -> times(+(encArg(x_17), encArg(x_27)), const) [0]
   encode_*(cons_*(x_18, x_28), +(x_111, x_211)) -> times(times(encArg(x_18), encArg(x_28)), +(encArg(x_111), encArg(x_211))) [0]
   encode_*(cons_*(x_18, x_28), cons_*(x_112, x_212)) -> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) [0]
   encode_*(cons_*(x_18, x_28), x_2) -> times(times(encArg(x_18), encArg(x_28)), const) [0]
   encode_*(x_1, +(x_113, x_213)) -> times(const, +(encArg(x_113), encArg(x_213))) [0]
   encode_*(x_1, cons_*(x_114, x_214)) -> times(const, times(encArg(x_114), encArg(x_214))) [0]
   encode_*(x_1, x_2) -> times(const, const) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> const [0]
   encode_*(v0, v1) -> const [0]
   encode_+(v0, v1) -> const [0]
   times(v0, v1) -> const [0]

The TRS has the following type information:
times :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
+ :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
encArg :: +:cons_*:const -> +:cons_*:const
cons_* :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
encode_* :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
encode_+ :: +:cons_*:const -> +:cons_*:const -> +:cons_*:const
const :: +:cons_*:const

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

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

   const => 0

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

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

   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: v0 >= 0, z' = v0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, x_2 >= 0, z'' = x_2, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: x_1 >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0, z' = x_1
   encode_*(z', z'') -{ 0 }-> times(0, 0) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: x_1 >= 0, x_113 >= 0, x_213 >= 0, z' = x_1, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, x_2 >= 0, z'' = x_2, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_+(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   times(z', z'') -{ 1 }-> 1 + times(x, y) + times(x, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z


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

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

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

   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 0) :|: z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z


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

   { times }
   { encArg }
   { encode_+ }
   { encode_* }

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

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

   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 0) :|: z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {times}, {encArg}, {encode_+}, {encode_*}

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

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

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

   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 0) :|: z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {times}, {encArg}, {encode_+}, {encode_*}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 0) :|: z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 0) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 0) :|: z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

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

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

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

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

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

(29) 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: z'

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

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

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

(31) 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: 26*z' + 8*z'^2

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> times(times(encArg(x_1''), encArg(x_2'')), times(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 }-> times(times(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 }-> times(times(encArg(x_1''), encArg(x_2'')), 1 + encArg(x_13) + encArg(x_23)) :|: x_1'' >= 0, x_13 >= 0, x_2'' >= 0, x_23 >= 0, z' = 1 + (1 + x_1'' + x_2'') + (1 + x_13 + x_23)
   encArg(z') -{ 0 }-> times(0, times(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 }-> times(0, 1 + encArg(x_15) + encArg(x_25)) :|: x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ 0 }-> times(1 + encArg(x_1') + encArg(x_2'), times(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 }-> times(1 + 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 }-> times(1 + encArg(x_1') + encArg(x_2'), 1 + 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), times(encArg(x_112), encArg(x_212))) :|: x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 0) :|: z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 0 }-> times(times(encArg(x_18), encArg(x_28)), 1 + encArg(x_111) + encArg(x_211)) :|: z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, times(encArg(x_114), encArg(x_214))) :|: z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> times(0, 1 + encArg(x_113) + encArg(x_213)) :|: z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), times(encArg(x_110), encArg(x_210))) :|: x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 0) :|: x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 0 }-> times(1 + encArg(x_17) + encArg(x_27), 1 + encArg(x_19) + encArg(x_29)) :|: z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {encode_+}, {encode_*}
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(n^1) [z'']
  encArg: runtime: O(n^2) [26*z' + 8*z'^2], size: O(n^1) [z']

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

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ s12 + s13 + 26*x_1' + 8*x_1'^2 + 26*x_12 + 8*x_12^2 + 26*x_2' + 8*x_2'^2 + 26*x_22 + 8*x_22^2 }-> s14 :|: s9 >= 0, s9 <= x_1', s10 >= 0, s10 <= x_2', s11 >= 0, s11 <= x_12, s12 >= 0, s12 <= x_22, s13 >= 0, s13 <= s12, s14 >= 0, s14 <= s13, 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') -{ 26*x_1' + 8*x_1'^2 + 26*x_2' + 8*x_2'^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_2', s17 >= 0, s17 <= 0, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s19 + s21 + s22 + 26*x_1'' + 8*x_1''^2 + 26*x_13 + 8*x_13^2 + 26*x_2'' + 8*x_2''^2 + 26*x_23 + 8*x_23^2 }-> s23 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_2'', s20 >= 0, s20 <= s19, s21 >= 0, s21 <= x_13, s22 >= 0, s22 <= x_23, s23 >= 0, s23 <= 1 + s21 + s22, 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') -{ s25 + s28 + s29 + 26*x_1'' + 8*x_1''^2 + 26*x_14 + 8*x_14^2 + 26*x_2'' + 8*x_2''^2 + 26*x_24 + 8*x_24^2 }-> s30 :|: s24 >= 0, s24 <= x_1'', s25 >= 0, s25 <= x_2'', s26 >= 0, s26 <= s25, s27 >= 0, s27 <= x_14, s28 >= 0, s28 <= x_24, s29 >= 0, s29 <= s28, s30 >= 0, s30 <= s29, 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') -{ s32 + 26*x_1'' + 8*x_1''^2 + 26*x_2'' + 8*x_2''^2 }-> s34 :|: s31 >= 0, s31 <= x_1'', s32 >= 0, s32 <= x_2'', s33 >= 0, s33 <= s32, s34 >= 0, s34 <= 0, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s35 + s36 + 26*x_15 + 8*x_15^2 + 26*x_25 + 8*x_25^2 }-> s37 :|: s35 >= 0, s35 <= x_15, s36 >= 0, s36 <= x_25, s37 >= 0, s37 <= 1 + s35 + s36, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ s39 + s40 + 26*x_16 + 8*x_16^2 + 26*x_26 + 8*x_26^2 }-> s41 :|: s38 >= 0, s38 <= x_16, s39 >= 0, s39 <= x_26, s40 >= 0, s40 <= s39, s41 >= 0, s41 <= s40, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 1 + s6 + s7 + 26*x_1' + 8*x_1'^2 + 26*x_11 + 8*x_11^2 + 26*x_2' + 8*x_2'^2 + 26*x_21 + 8*x_21^2 }-> s8 :|: s4 >= 0, s4 <= x_1', s5 >= 0, s5 <= x_2', s6 >= 0, s6 <= x_11, s7 >= 0, s7 <= x_21, s8 >= 0, s8 <= 1 + s6 + s7, 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 26*x_1 + 8*x_1^2 + 26*x_2 + 8*x_2^2 }-> 1 + s2 + s3 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 1 + s44 + s45 + 26*x_17 + 8*x_17^2 + 26*x_19 + 8*x_19^2 + 26*x_27 + 8*x_27^2 + 26*x_29 + 8*x_29^2 }-> s46 :|: s42 >= 0, s42 <= x_17, s43 >= 0, s43 <= x_27, s44 >= 0, s44 <= x_19, s45 >= 0, s45 <= x_29, s46 >= 0, s46 <= 1 + s44 + s45, z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ s50 + s51 + 26*x_110 + 8*x_110^2 + 26*x_17 + 8*x_17^2 + 26*x_210 + 8*x_210^2 + 26*x_27 + 8*x_27^2 }-> s52 :|: s47 >= 0, s47 <= x_17, s48 >= 0, s48 <= x_27, s49 >= 0, s49 <= x_110, s50 >= 0, s50 <= x_210, s51 >= 0, s51 <= s50, s52 >= 0, s52 <= s51, x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 26*x_17 + 8*x_17^2 + 26*x_27 + 8*x_27^2 }-> s55 :|: s53 >= 0, s53 <= x_17, s54 >= 0, s54 <= x_27, s55 >= 0, s55 <= 0, x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 1 + s57 + s59 + s60 + 26*x_111 + 8*x_111^2 + 26*x_18 + 8*x_18^2 + 26*x_211 + 8*x_211^2 + 26*x_28 + 8*x_28^2 }-> s61 :|: s56 >= 0, s56 <= x_18, s57 >= 0, s57 <= x_28, s58 >= 0, s58 <= s57, s59 >= 0, s59 <= x_111, s60 >= 0, s60 <= x_211, s61 >= 0, s61 <= 1 + s59 + s60, z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ s63 + s66 + s67 + 26*x_112 + 8*x_112^2 + 26*x_18 + 8*x_18^2 + 26*x_212 + 8*x_212^2 + 26*x_28 + 8*x_28^2 }-> s68 :|: s62 >= 0, s62 <= x_18, s63 >= 0, s63 <= x_28, s64 >= 0, s64 <= s63, s65 >= 0, s65 <= x_112, s66 >= 0, s66 <= x_212, s67 >= 0, s67 <= s66, s68 >= 0, s68 <= s67, x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ s70 + 26*x_18 + 8*x_18^2 + 26*x_28 + 8*x_28^2 }-> s72 :|: s69 >= 0, s69 <= x_18, s70 >= 0, s70 <= x_28, s71 >= 0, s71 <= s70, s72 >= 0, s72 <= 0, z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 1 + s73 + s74 + 26*x_113 + 8*x_113^2 + 26*x_213 + 8*x_213^2 }-> s75 :|: s73 >= 0, s73 <= x_113, s74 >= 0, s74 <= x_213, s75 >= 0, s75 <= 1 + s73 + s74, z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ s77 + s78 + 26*x_114 + 8*x_114^2 + 26*x_214 + 8*x_214^2 }-> s79 :|: s76 >= 0, s76 <= x_114, s77 >= 0, s77 <= x_214, s78 >= 0, s78 <= s77, s79 >= 0, s79 <= s78, z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 26*z' + 8*z'^2 + 26*z'' + 8*z''^2 }-> 1 + s80 + s81 :|: s80 >= 0, s80 <= z', s81 >= 0, s81 <= z'', z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {encode_+}, {encode_*}
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(n^1) [z'']
  encArg: runtime: O(n^2) [26*z' + 8*z'^2], size: O(n^1) [z']

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

(35) 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: 1 + z' + z''

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ s12 + s13 + 26*x_1' + 8*x_1'^2 + 26*x_12 + 8*x_12^2 + 26*x_2' + 8*x_2'^2 + 26*x_22 + 8*x_22^2 }-> s14 :|: s9 >= 0, s9 <= x_1', s10 >= 0, s10 <= x_2', s11 >= 0, s11 <= x_12, s12 >= 0, s12 <= x_22, s13 >= 0, s13 <= s12, s14 >= 0, s14 <= s13, 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') -{ 26*x_1' + 8*x_1'^2 + 26*x_2' + 8*x_2'^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_2', s17 >= 0, s17 <= 0, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s19 + s21 + s22 + 26*x_1'' + 8*x_1''^2 + 26*x_13 + 8*x_13^2 + 26*x_2'' + 8*x_2''^2 + 26*x_23 + 8*x_23^2 }-> s23 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_2'', s20 >= 0, s20 <= s19, s21 >= 0, s21 <= x_13, s22 >= 0, s22 <= x_23, s23 >= 0, s23 <= 1 + s21 + s22, 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') -{ s25 + s28 + s29 + 26*x_1'' + 8*x_1''^2 + 26*x_14 + 8*x_14^2 + 26*x_2'' + 8*x_2''^2 + 26*x_24 + 8*x_24^2 }-> s30 :|: s24 >= 0, s24 <= x_1'', s25 >= 0, s25 <= x_2'', s26 >= 0, s26 <= s25, s27 >= 0, s27 <= x_14, s28 >= 0, s28 <= x_24, s29 >= 0, s29 <= s28, s30 >= 0, s30 <= s29, 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') -{ s32 + 26*x_1'' + 8*x_1''^2 + 26*x_2'' + 8*x_2''^2 }-> s34 :|: s31 >= 0, s31 <= x_1'', s32 >= 0, s32 <= x_2'', s33 >= 0, s33 <= s32, s34 >= 0, s34 <= 0, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s35 + s36 + 26*x_15 + 8*x_15^2 + 26*x_25 + 8*x_25^2 }-> s37 :|: s35 >= 0, s35 <= x_15, s36 >= 0, s36 <= x_25, s37 >= 0, s37 <= 1 + s35 + s36, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ s39 + s40 + 26*x_16 + 8*x_16^2 + 26*x_26 + 8*x_26^2 }-> s41 :|: s38 >= 0, s38 <= x_16, s39 >= 0, s39 <= x_26, s40 >= 0, s40 <= s39, s41 >= 0, s41 <= s40, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 1 + s6 + s7 + 26*x_1' + 8*x_1'^2 + 26*x_11 + 8*x_11^2 + 26*x_2' + 8*x_2'^2 + 26*x_21 + 8*x_21^2 }-> s8 :|: s4 >= 0, s4 <= x_1', s5 >= 0, s5 <= x_2', s6 >= 0, s6 <= x_11, s7 >= 0, s7 <= x_21, s8 >= 0, s8 <= 1 + s6 + s7, 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 26*x_1 + 8*x_1^2 + 26*x_2 + 8*x_2^2 }-> 1 + s2 + s3 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 1 + s44 + s45 + 26*x_17 + 8*x_17^2 + 26*x_19 + 8*x_19^2 + 26*x_27 + 8*x_27^2 + 26*x_29 + 8*x_29^2 }-> s46 :|: s42 >= 0, s42 <= x_17, s43 >= 0, s43 <= x_27, s44 >= 0, s44 <= x_19, s45 >= 0, s45 <= x_29, s46 >= 0, s46 <= 1 + s44 + s45, z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ s50 + s51 + 26*x_110 + 8*x_110^2 + 26*x_17 + 8*x_17^2 + 26*x_210 + 8*x_210^2 + 26*x_27 + 8*x_27^2 }-> s52 :|: s47 >= 0, s47 <= x_17, s48 >= 0, s48 <= x_27, s49 >= 0, s49 <= x_110, s50 >= 0, s50 <= x_210, s51 >= 0, s51 <= s50, s52 >= 0, s52 <= s51, x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 26*x_17 + 8*x_17^2 + 26*x_27 + 8*x_27^2 }-> s55 :|: s53 >= 0, s53 <= x_17, s54 >= 0, s54 <= x_27, s55 >= 0, s55 <= 0, x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 1 + s57 + s59 + s60 + 26*x_111 + 8*x_111^2 + 26*x_18 + 8*x_18^2 + 26*x_211 + 8*x_211^2 + 26*x_28 + 8*x_28^2 }-> s61 :|: s56 >= 0, s56 <= x_18, s57 >= 0, s57 <= x_28, s58 >= 0, s58 <= s57, s59 >= 0, s59 <= x_111, s60 >= 0, s60 <= x_211, s61 >= 0, s61 <= 1 + s59 + s60, z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ s63 + s66 + s67 + 26*x_112 + 8*x_112^2 + 26*x_18 + 8*x_18^2 + 26*x_212 + 8*x_212^2 + 26*x_28 + 8*x_28^2 }-> s68 :|: s62 >= 0, s62 <= x_18, s63 >= 0, s63 <= x_28, s64 >= 0, s64 <= s63, s65 >= 0, s65 <= x_112, s66 >= 0, s66 <= x_212, s67 >= 0, s67 <= s66, s68 >= 0, s68 <= s67, x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ s70 + 26*x_18 + 8*x_18^2 + 26*x_28 + 8*x_28^2 }-> s72 :|: s69 >= 0, s69 <= x_18, s70 >= 0, s70 <= x_28, s71 >= 0, s71 <= s70, s72 >= 0, s72 <= 0, z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 1 + s73 + s74 + 26*x_113 + 8*x_113^2 + 26*x_213 + 8*x_213^2 }-> s75 :|: s73 >= 0, s73 <= x_113, s74 >= 0, s74 <= x_213, s75 >= 0, s75 <= 1 + s73 + s74, z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ s77 + s78 + 26*x_114 + 8*x_114^2 + 26*x_214 + 8*x_214^2 }-> s79 :|: s76 >= 0, s76 <= x_114, s77 >= 0, s77 <= x_214, s78 >= 0, s78 <= s77, s79 >= 0, s79 <= s78, z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 26*z' + 8*z'^2 + 26*z'' + 8*z''^2 }-> 1 + s80 + s81 :|: s80 >= 0, s80 <= z', s81 >= 0, s81 <= z'', z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {encode_+}, {encode_*}
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(n^1) [z'']
  encArg: runtime: O(n^2) [26*z' + 8*z'^2], size: O(n^1) [z']
  encode_+: runtime: ?, size: O(n^1) [1 + z' + z'']

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

(37) 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: 26*z' + 8*z'^2 + 26*z'' + 8*z''^2

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ s12 + s13 + 26*x_1' + 8*x_1'^2 + 26*x_12 + 8*x_12^2 + 26*x_2' + 8*x_2'^2 + 26*x_22 + 8*x_22^2 }-> s14 :|: s9 >= 0, s9 <= x_1', s10 >= 0, s10 <= x_2', s11 >= 0, s11 <= x_12, s12 >= 0, s12 <= x_22, s13 >= 0, s13 <= s12, s14 >= 0, s14 <= s13, 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') -{ 26*x_1' + 8*x_1'^2 + 26*x_2' + 8*x_2'^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_2', s17 >= 0, s17 <= 0, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s19 + s21 + s22 + 26*x_1'' + 8*x_1''^2 + 26*x_13 + 8*x_13^2 + 26*x_2'' + 8*x_2''^2 + 26*x_23 + 8*x_23^2 }-> s23 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_2'', s20 >= 0, s20 <= s19, s21 >= 0, s21 <= x_13, s22 >= 0, s22 <= x_23, s23 >= 0, s23 <= 1 + s21 + s22, 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') -{ s25 + s28 + s29 + 26*x_1'' + 8*x_1''^2 + 26*x_14 + 8*x_14^2 + 26*x_2'' + 8*x_2''^2 + 26*x_24 + 8*x_24^2 }-> s30 :|: s24 >= 0, s24 <= x_1'', s25 >= 0, s25 <= x_2'', s26 >= 0, s26 <= s25, s27 >= 0, s27 <= x_14, s28 >= 0, s28 <= x_24, s29 >= 0, s29 <= s28, s30 >= 0, s30 <= s29, 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') -{ s32 + 26*x_1'' + 8*x_1''^2 + 26*x_2'' + 8*x_2''^2 }-> s34 :|: s31 >= 0, s31 <= x_1'', s32 >= 0, s32 <= x_2'', s33 >= 0, s33 <= s32, s34 >= 0, s34 <= 0, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s35 + s36 + 26*x_15 + 8*x_15^2 + 26*x_25 + 8*x_25^2 }-> s37 :|: s35 >= 0, s35 <= x_15, s36 >= 0, s36 <= x_25, s37 >= 0, s37 <= 1 + s35 + s36, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ s39 + s40 + 26*x_16 + 8*x_16^2 + 26*x_26 + 8*x_26^2 }-> s41 :|: s38 >= 0, s38 <= x_16, s39 >= 0, s39 <= x_26, s40 >= 0, s40 <= s39, s41 >= 0, s41 <= s40, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 1 + s6 + s7 + 26*x_1' + 8*x_1'^2 + 26*x_11 + 8*x_11^2 + 26*x_2' + 8*x_2'^2 + 26*x_21 + 8*x_21^2 }-> s8 :|: s4 >= 0, s4 <= x_1', s5 >= 0, s5 <= x_2', s6 >= 0, s6 <= x_11, s7 >= 0, s7 <= x_21, s8 >= 0, s8 <= 1 + s6 + s7, 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 26*x_1 + 8*x_1^2 + 26*x_2 + 8*x_2^2 }-> 1 + s2 + s3 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 1 + s44 + s45 + 26*x_17 + 8*x_17^2 + 26*x_19 + 8*x_19^2 + 26*x_27 + 8*x_27^2 + 26*x_29 + 8*x_29^2 }-> s46 :|: s42 >= 0, s42 <= x_17, s43 >= 0, s43 <= x_27, s44 >= 0, s44 <= x_19, s45 >= 0, s45 <= x_29, s46 >= 0, s46 <= 1 + s44 + s45, z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ s50 + s51 + 26*x_110 + 8*x_110^2 + 26*x_17 + 8*x_17^2 + 26*x_210 + 8*x_210^2 + 26*x_27 + 8*x_27^2 }-> s52 :|: s47 >= 0, s47 <= x_17, s48 >= 0, s48 <= x_27, s49 >= 0, s49 <= x_110, s50 >= 0, s50 <= x_210, s51 >= 0, s51 <= s50, s52 >= 0, s52 <= s51, x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 26*x_17 + 8*x_17^2 + 26*x_27 + 8*x_27^2 }-> s55 :|: s53 >= 0, s53 <= x_17, s54 >= 0, s54 <= x_27, s55 >= 0, s55 <= 0, x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 1 + s57 + s59 + s60 + 26*x_111 + 8*x_111^2 + 26*x_18 + 8*x_18^2 + 26*x_211 + 8*x_211^2 + 26*x_28 + 8*x_28^2 }-> s61 :|: s56 >= 0, s56 <= x_18, s57 >= 0, s57 <= x_28, s58 >= 0, s58 <= s57, s59 >= 0, s59 <= x_111, s60 >= 0, s60 <= x_211, s61 >= 0, s61 <= 1 + s59 + s60, z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ s63 + s66 + s67 + 26*x_112 + 8*x_112^2 + 26*x_18 + 8*x_18^2 + 26*x_212 + 8*x_212^2 + 26*x_28 + 8*x_28^2 }-> s68 :|: s62 >= 0, s62 <= x_18, s63 >= 0, s63 <= x_28, s64 >= 0, s64 <= s63, s65 >= 0, s65 <= x_112, s66 >= 0, s66 <= x_212, s67 >= 0, s67 <= s66, s68 >= 0, s68 <= s67, x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ s70 + 26*x_18 + 8*x_18^2 + 26*x_28 + 8*x_28^2 }-> s72 :|: s69 >= 0, s69 <= x_18, s70 >= 0, s70 <= x_28, s71 >= 0, s71 <= s70, s72 >= 0, s72 <= 0, z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 1 + s73 + s74 + 26*x_113 + 8*x_113^2 + 26*x_213 + 8*x_213^2 }-> s75 :|: s73 >= 0, s73 <= x_113, s74 >= 0, s74 <= x_213, s75 >= 0, s75 <= 1 + s73 + s74, z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ s77 + s78 + 26*x_114 + 8*x_114^2 + 26*x_214 + 8*x_214^2 }-> s79 :|: s76 >= 0, s76 <= x_114, s77 >= 0, s77 <= x_214, s78 >= 0, s78 <= s77, s79 >= 0, s79 <= s78, z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 26*z' + 8*z'^2 + 26*z'' + 8*z''^2 }-> 1 + s80 + s81 :|: s80 >= 0, s80 <= z', s81 >= 0, s81 <= z'', z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {encode_*}
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(n^1) [z'']
  encArg: runtime: O(n^2) [26*z' + 8*z'^2], size: O(n^1) [z']
  encode_+: runtime: O(n^2) [26*z' + 8*z'^2 + 26*z'' + 8*z''^2], size: O(n^1) [1 + z' + z'']

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ s12 + s13 + 26*x_1' + 8*x_1'^2 + 26*x_12 + 8*x_12^2 + 26*x_2' + 8*x_2'^2 + 26*x_22 + 8*x_22^2 }-> s14 :|: s9 >= 0, s9 <= x_1', s10 >= 0, s10 <= x_2', s11 >= 0, s11 <= x_12, s12 >= 0, s12 <= x_22, s13 >= 0, s13 <= s12, s14 >= 0, s14 <= s13, 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') -{ 26*x_1' + 8*x_1'^2 + 26*x_2' + 8*x_2'^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_2', s17 >= 0, s17 <= 0, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s19 + s21 + s22 + 26*x_1'' + 8*x_1''^2 + 26*x_13 + 8*x_13^2 + 26*x_2'' + 8*x_2''^2 + 26*x_23 + 8*x_23^2 }-> s23 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_2'', s20 >= 0, s20 <= s19, s21 >= 0, s21 <= x_13, s22 >= 0, s22 <= x_23, s23 >= 0, s23 <= 1 + s21 + s22, 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') -{ s25 + s28 + s29 + 26*x_1'' + 8*x_1''^2 + 26*x_14 + 8*x_14^2 + 26*x_2'' + 8*x_2''^2 + 26*x_24 + 8*x_24^2 }-> s30 :|: s24 >= 0, s24 <= x_1'', s25 >= 0, s25 <= x_2'', s26 >= 0, s26 <= s25, s27 >= 0, s27 <= x_14, s28 >= 0, s28 <= x_24, s29 >= 0, s29 <= s28, s30 >= 0, s30 <= s29, 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') -{ s32 + 26*x_1'' + 8*x_1''^2 + 26*x_2'' + 8*x_2''^2 }-> s34 :|: s31 >= 0, s31 <= x_1'', s32 >= 0, s32 <= x_2'', s33 >= 0, s33 <= s32, s34 >= 0, s34 <= 0, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s35 + s36 + 26*x_15 + 8*x_15^2 + 26*x_25 + 8*x_25^2 }-> s37 :|: s35 >= 0, s35 <= x_15, s36 >= 0, s36 <= x_25, s37 >= 0, s37 <= 1 + s35 + s36, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ s39 + s40 + 26*x_16 + 8*x_16^2 + 26*x_26 + 8*x_26^2 }-> s41 :|: s38 >= 0, s38 <= x_16, s39 >= 0, s39 <= x_26, s40 >= 0, s40 <= s39, s41 >= 0, s41 <= s40, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 1 + s6 + s7 + 26*x_1' + 8*x_1'^2 + 26*x_11 + 8*x_11^2 + 26*x_2' + 8*x_2'^2 + 26*x_21 + 8*x_21^2 }-> s8 :|: s4 >= 0, s4 <= x_1', s5 >= 0, s5 <= x_2', s6 >= 0, s6 <= x_11, s7 >= 0, s7 <= x_21, s8 >= 0, s8 <= 1 + s6 + s7, 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 26*x_1 + 8*x_1^2 + 26*x_2 + 8*x_2^2 }-> 1 + s2 + s3 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 1 + s44 + s45 + 26*x_17 + 8*x_17^2 + 26*x_19 + 8*x_19^2 + 26*x_27 + 8*x_27^2 + 26*x_29 + 8*x_29^2 }-> s46 :|: s42 >= 0, s42 <= x_17, s43 >= 0, s43 <= x_27, s44 >= 0, s44 <= x_19, s45 >= 0, s45 <= x_29, s46 >= 0, s46 <= 1 + s44 + s45, z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ s50 + s51 + 26*x_110 + 8*x_110^2 + 26*x_17 + 8*x_17^2 + 26*x_210 + 8*x_210^2 + 26*x_27 + 8*x_27^2 }-> s52 :|: s47 >= 0, s47 <= x_17, s48 >= 0, s48 <= x_27, s49 >= 0, s49 <= x_110, s50 >= 0, s50 <= x_210, s51 >= 0, s51 <= s50, s52 >= 0, s52 <= s51, x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 26*x_17 + 8*x_17^2 + 26*x_27 + 8*x_27^2 }-> s55 :|: s53 >= 0, s53 <= x_17, s54 >= 0, s54 <= x_27, s55 >= 0, s55 <= 0, x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 1 + s57 + s59 + s60 + 26*x_111 + 8*x_111^2 + 26*x_18 + 8*x_18^2 + 26*x_211 + 8*x_211^2 + 26*x_28 + 8*x_28^2 }-> s61 :|: s56 >= 0, s56 <= x_18, s57 >= 0, s57 <= x_28, s58 >= 0, s58 <= s57, s59 >= 0, s59 <= x_111, s60 >= 0, s60 <= x_211, s61 >= 0, s61 <= 1 + s59 + s60, z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ s63 + s66 + s67 + 26*x_112 + 8*x_112^2 + 26*x_18 + 8*x_18^2 + 26*x_212 + 8*x_212^2 + 26*x_28 + 8*x_28^2 }-> s68 :|: s62 >= 0, s62 <= x_18, s63 >= 0, s63 <= x_28, s64 >= 0, s64 <= s63, s65 >= 0, s65 <= x_112, s66 >= 0, s66 <= x_212, s67 >= 0, s67 <= s66, s68 >= 0, s68 <= s67, x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ s70 + 26*x_18 + 8*x_18^2 + 26*x_28 + 8*x_28^2 }-> s72 :|: s69 >= 0, s69 <= x_18, s70 >= 0, s70 <= x_28, s71 >= 0, s71 <= s70, s72 >= 0, s72 <= 0, z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 1 + s73 + s74 + 26*x_113 + 8*x_113^2 + 26*x_213 + 8*x_213^2 }-> s75 :|: s73 >= 0, s73 <= x_113, s74 >= 0, s74 <= x_213, s75 >= 0, s75 <= 1 + s73 + s74, z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ s77 + s78 + 26*x_114 + 8*x_114^2 + 26*x_214 + 8*x_214^2 }-> s79 :|: s76 >= 0, s76 <= x_114, s77 >= 0, s77 <= x_214, s78 >= 0, s78 <= s77, s79 >= 0, s79 <= s78, z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 26*z' + 8*z'^2 + 26*z'' + 8*z''^2 }-> 1 + s80 + s81 :|: s80 >= 0, s80 <= z', s81 >= 0, s81 <= z'', z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {encode_*}
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(n^1) [z'']
  encArg: runtime: O(n^2) [26*z' + 8*z'^2], size: O(n^1) [z']
  encode_+: runtime: O(n^2) [26*z' + 8*z'^2 + 26*z'' + 8*z''^2], size: O(n^1) [1 + z' + z'']

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

(41) 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: z''

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ s12 + s13 + 26*x_1' + 8*x_1'^2 + 26*x_12 + 8*x_12^2 + 26*x_2' + 8*x_2'^2 + 26*x_22 + 8*x_22^2 }-> s14 :|: s9 >= 0, s9 <= x_1', s10 >= 0, s10 <= x_2', s11 >= 0, s11 <= x_12, s12 >= 0, s12 <= x_22, s13 >= 0, s13 <= s12, s14 >= 0, s14 <= s13, 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') -{ 26*x_1' + 8*x_1'^2 + 26*x_2' + 8*x_2'^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_2', s17 >= 0, s17 <= 0, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s19 + s21 + s22 + 26*x_1'' + 8*x_1''^2 + 26*x_13 + 8*x_13^2 + 26*x_2'' + 8*x_2''^2 + 26*x_23 + 8*x_23^2 }-> s23 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_2'', s20 >= 0, s20 <= s19, s21 >= 0, s21 <= x_13, s22 >= 0, s22 <= x_23, s23 >= 0, s23 <= 1 + s21 + s22, 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') -{ s25 + s28 + s29 + 26*x_1'' + 8*x_1''^2 + 26*x_14 + 8*x_14^2 + 26*x_2'' + 8*x_2''^2 + 26*x_24 + 8*x_24^2 }-> s30 :|: s24 >= 0, s24 <= x_1'', s25 >= 0, s25 <= x_2'', s26 >= 0, s26 <= s25, s27 >= 0, s27 <= x_14, s28 >= 0, s28 <= x_24, s29 >= 0, s29 <= s28, s30 >= 0, s30 <= s29, 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') -{ s32 + 26*x_1'' + 8*x_1''^2 + 26*x_2'' + 8*x_2''^2 }-> s34 :|: s31 >= 0, s31 <= x_1'', s32 >= 0, s32 <= x_2'', s33 >= 0, s33 <= s32, s34 >= 0, s34 <= 0, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s35 + s36 + 26*x_15 + 8*x_15^2 + 26*x_25 + 8*x_25^2 }-> s37 :|: s35 >= 0, s35 <= x_15, s36 >= 0, s36 <= x_25, s37 >= 0, s37 <= 1 + s35 + s36, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ s39 + s40 + 26*x_16 + 8*x_16^2 + 26*x_26 + 8*x_26^2 }-> s41 :|: s38 >= 0, s38 <= x_16, s39 >= 0, s39 <= x_26, s40 >= 0, s40 <= s39, s41 >= 0, s41 <= s40, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 1 + s6 + s7 + 26*x_1' + 8*x_1'^2 + 26*x_11 + 8*x_11^2 + 26*x_2' + 8*x_2'^2 + 26*x_21 + 8*x_21^2 }-> s8 :|: s4 >= 0, s4 <= x_1', s5 >= 0, s5 <= x_2', s6 >= 0, s6 <= x_11, s7 >= 0, s7 <= x_21, s8 >= 0, s8 <= 1 + s6 + s7, 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 26*x_1 + 8*x_1^2 + 26*x_2 + 8*x_2^2 }-> 1 + s2 + s3 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 1 + s44 + s45 + 26*x_17 + 8*x_17^2 + 26*x_19 + 8*x_19^2 + 26*x_27 + 8*x_27^2 + 26*x_29 + 8*x_29^2 }-> s46 :|: s42 >= 0, s42 <= x_17, s43 >= 0, s43 <= x_27, s44 >= 0, s44 <= x_19, s45 >= 0, s45 <= x_29, s46 >= 0, s46 <= 1 + s44 + s45, z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ s50 + s51 + 26*x_110 + 8*x_110^2 + 26*x_17 + 8*x_17^2 + 26*x_210 + 8*x_210^2 + 26*x_27 + 8*x_27^2 }-> s52 :|: s47 >= 0, s47 <= x_17, s48 >= 0, s48 <= x_27, s49 >= 0, s49 <= x_110, s50 >= 0, s50 <= x_210, s51 >= 0, s51 <= s50, s52 >= 0, s52 <= s51, x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 26*x_17 + 8*x_17^2 + 26*x_27 + 8*x_27^2 }-> s55 :|: s53 >= 0, s53 <= x_17, s54 >= 0, s54 <= x_27, s55 >= 0, s55 <= 0, x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 1 + s57 + s59 + s60 + 26*x_111 + 8*x_111^2 + 26*x_18 + 8*x_18^2 + 26*x_211 + 8*x_211^2 + 26*x_28 + 8*x_28^2 }-> s61 :|: s56 >= 0, s56 <= x_18, s57 >= 0, s57 <= x_28, s58 >= 0, s58 <= s57, s59 >= 0, s59 <= x_111, s60 >= 0, s60 <= x_211, s61 >= 0, s61 <= 1 + s59 + s60, z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ s63 + s66 + s67 + 26*x_112 + 8*x_112^2 + 26*x_18 + 8*x_18^2 + 26*x_212 + 8*x_212^2 + 26*x_28 + 8*x_28^2 }-> s68 :|: s62 >= 0, s62 <= x_18, s63 >= 0, s63 <= x_28, s64 >= 0, s64 <= s63, s65 >= 0, s65 <= x_112, s66 >= 0, s66 <= x_212, s67 >= 0, s67 <= s66, s68 >= 0, s68 <= s67, x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ s70 + 26*x_18 + 8*x_18^2 + 26*x_28 + 8*x_28^2 }-> s72 :|: s69 >= 0, s69 <= x_18, s70 >= 0, s70 <= x_28, s71 >= 0, s71 <= s70, s72 >= 0, s72 <= 0, z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 1 + s73 + s74 + 26*x_113 + 8*x_113^2 + 26*x_213 + 8*x_213^2 }-> s75 :|: s73 >= 0, s73 <= x_113, s74 >= 0, s74 <= x_213, s75 >= 0, s75 <= 1 + s73 + s74, z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ s77 + s78 + 26*x_114 + 8*x_114^2 + 26*x_214 + 8*x_214^2 }-> s79 :|: s76 >= 0, s76 <= x_114, s77 >= 0, s77 <= x_214, s78 >= 0, s78 <= s77, s79 >= 0, s79 <= s78, z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 26*z' + 8*z'^2 + 26*z'' + 8*z''^2 }-> 1 + s80 + s81 :|: s80 >= 0, s80 <= z', s81 >= 0, s81 <= z'', z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {encode_*}
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(n^1) [z'']
  encArg: runtime: O(n^2) [26*z' + 8*z'^2], size: O(n^1) [z']
  encode_+: runtime: O(n^2) [26*z' + 8*z'^2 + 26*z'' + 8*z''^2], size: O(n^1) [1 + z' + z'']
  encode_*: runtime: ?, size: O(n^1) [z'']

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

(43) 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: 3 + 315*z' + 96*z'^2 + 324*z'' + 96*z''^2

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

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

   encArg(z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ s12 + s13 + 26*x_1' + 8*x_1'^2 + 26*x_12 + 8*x_12^2 + 26*x_2' + 8*x_2'^2 + 26*x_22 + 8*x_22^2 }-> s14 :|: s9 >= 0, s9 <= x_1', s10 >= 0, s10 <= x_2', s11 >= 0, s11 <= x_12, s12 >= 0, s12 <= x_22, s13 >= 0, s13 <= s12, s14 >= 0, s14 <= s13, 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') -{ 26*x_1' + 8*x_1'^2 + 26*x_2' + 8*x_2'^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_2', s17 >= 0, s17 <= 0, x_2' >= 0, z' = 1 + (1 + x_1' + x_2') + x_2, x_1' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s19 + s21 + s22 + 26*x_1'' + 8*x_1''^2 + 26*x_13 + 8*x_13^2 + 26*x_2'' + 8*x_2''^2 + 26*x_23 + 8*x_23^2 }-> s23 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_2'', s20 >= 0, s20 <= s19, s21 >= 0, s21 <= x_13, s22 >= 0, s22 <= x_23, s23 >= 0, s23 <= 1 + s21 + s22, 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') -{ s25 + s28 + s29 + 26*x_1'' + 8*x_1''^2 + 26*x_14 + 8*x_14^2 + 26*x_2'' + 8*x_2''^2 + 26*x_24 + 8*x_24^2 }-> s30 :|: s24 >= 0, s24 <= x_1'', s25 >= 0, s25 <= x_2'', s26 >= 0, s26 <= s25, s27 >= 0, s27 <= x_14, s28 >= 0, s28 <= x_24, s29 >= 0, s29 <= s28, s30 >= 0, s30 <= s29, 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') -{ s32 + 26*x_1'' + 8*x_1''^2 + 26*x_2'' + 8*x_2''^2 }-> s34 :|: s31 >= 0, s31 <= x_1'', s32 >= 0, s32 <= x_2'', s33 >= 0, s33 <= s32, s34 >= 0, s34 <= 0, z' = 1 + (1 + x_1'' + x_2'') + x_2, x_1'' >= 0, x_2'' >= 0, x_2 >= 0
   encArg(z') -{ 1 + s35 + s36 + 26*x_15 + 8*x_15^2 + 26*x_25 + 8*x_25^2 }-> s37 :|: s35 >= 0, s35 <= x_15, s36 >= 0, s36 <= x_25, s37 >= 0, s37 <= 1 + s35 + s36, x_15 >= 0, x_1 >= 0, x_25 >= 0, z' = 1 + x_1 + (1 + x_15 + x_25)
   encArg(z') -{ s39 + s40 + 26*x_16 + 8*x_16^2 + 26*x_26 + 8*x_26^2 }-> s41 :|: s38 >= 0, s38 <= x_16, s39 >= 0, s39 <= x_26, s40 >= 0, s40 <= s39, s41 >= 0, s41 <= s40, x_1 >= 0, x_16 >= 0, x_26 >= 0, z' = 1 + x_1 + (1 + x_16 + x_26)
   encArg(z') -{ 1 + s6 + s7 + 26*x_1' + 8*x_1'^2 + 26*x_11 + 8*x_11^2 + 26*x_2' + 8*x_2'^2 + 26*x_21 + 8*x_21^2 }-> s8 :|: s4 >= 0, s4 <= x_1', s5 >= 0, s5 <= x_2', s6 >= 0, s6 <= x_11, s7 >= 0, s7 <= x_21, s8 >= 0, s8 <= 1 + s6 + s7, 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 }-> 0 :|: z' >= 0
   encArg(z') -{ 26*x_1 + 8*x_1^2 + 26*x_2 + 8*x_2^2 }-> 1 + s2 + s3 :|: s2 >= 0, s2 <= x_1, s3 >= 0, s3 <= x_2, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_*(z', z'') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z'' >= 0
   encode_*(z', z'') -{ 1 + s44 + s45 + 26*x_17 + 8*x_17^2 + 26*x_19 + 8*x_19^2 + 26*x_27 + 8*x_27^2 + 26*x_29 + 8*x_29^2 }-> s46 :|: s42 >= 0, s42 <= x_17, s43 >= 0, s43 <= x_27, s44 >= 0, s44 <= x_19, s45 >= 0, s45 <= x_29, s46 >= 0, s46 <= 1 + s44 + s45, z'' = 1 + x_19 + x_29, x_17 >= 0, x_27 >= 0, z' = 1 + x_17 + x_27, x_19 >= 0, x_29 >= 0
   encode_*(z', z'') -{ s50 + s51 + 26*x_110 + 8*x_110^2 + 26*x_17 + 8*x_17^2 + 26*x_210 + 8*x_210^2 + 26*x_27 + 8*x_27^2 }-> s52 :|: s47 >= 0, s47 <= x_17, s48 >= 0, s48 <= x_27, s49 >= 0, s49 <= x_110, s50 >= 0, s50 <= x_210, s51 >= 0, s51 <= s50, s52 >= 0, s52 <= s51, x_17 >= 0, x_27 >= 0, z'' = 1 + x_110 + x_210, x_110 >= 0, z' = 1 + x_17 + x_27, x_210 >= 0
   encode_*(z', z'') -{ 26*x_17 + 8*x_17^2 + 26*x_27 + 8*x_27^2 }-> s55 :|: s53 >= 0, s53 <= x_17, s54 >= 0, s54 <= x_27, s55 >= 0, s55 <= 0, x_17 >= 0, x_27 >= 0, z'' >= 0, z' = 1 + x_17 + x_27
   encode_*(z', z'') -{ 1 + s57 + s59 + s60 + 26*x_111 + 8*x_111^2 + 26*x_18 + 8*x_18^2 + 26*x_211 + 8*x_211^2 + 26*x_28 + 8*x_28^2 }-> s61 :|: s56 >= 0, s56 <= x_18, s57 >= 0, s57 <= x_28, s58 >= 0, s58 <= s57, s59 >= 0, s59 <= x_111, s60 >= 0, s60 <= x_211, s61 >= 0, s61 <= 1 + s59 + s60, z'' = 1 + x_111 + x_211, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_211 >= 0, x_111 >= 0
   encode_*(z', z'') -{ s63 + s66 + s67 + 26*x_112 + 8*x_112^2 + 26*x_18 + 8*x_18^2 + 26*x_212 + 8*x_212^2 + 26*x_28 + 8*x_28^2 }-> s68 :|: s62 >= 0, s62 <= x_18, s63 >= 0, s63 <= x_28, s64 >= 0, s64 <= s63, s65 >= 0, s65 <= x_112, s66 >= 0, s66 <= x_212, s67 >= 0, s67 <= s66, s68 >= 0, s68 <= s67, x_212 >= 0, z'' = 1 + x_112 + x_212, z' = 1 + x_18 + x_28, x_18 >= 0, x_28 >= 0, x_112 >= 0
   encode_*(z', z'') -{ s70 + 26*x_18 + 8*x_18^2 + 26*x_28 + 8*x_28^2 }-> s72 :|: s69 >= 0, s69 <= x_18, s70 >= 0, s70 <= x_28, s71 >= 0, s71 <= s70, s72 >= 0, s72 <= 0, z' = 1 + x_18 + x_28, z'' >= 0, x_18 >= 0, x_28 >= 0
   encode_*(z', z'') -{ 1 + s73 + s74 + 26*x_113 + 8*x_113^2 + 26*x_213 + 8*x_213^2 }-> s75 :|: s73 >= 0, s73 <= x_113, s74 >= 0, s74 <= x_213, s75 >= 0, s75 <= 1 + s73 + s74, z' >= 0, x_113 >= 0, x_213 >= 0, z'' = 1 + x_113 + x_213
   encode_*(z', z'') -{ s77 + s78 + 26*x_114 + 8*x_114^2 + 26*x_214 + 8*x_214^2 }-> s79 :|: s76 >= 0, s76 <= x_114, s77 >= 0, s77 <= x_214, s78 >= 0, s78 <= s77, s79 >= 0, s79 <= s78, z' >= 0, z'' = 1 + x_114 + x_214, x_114 >= 0, x_214 >= 0
   encode_*(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   encode_+(z', z'') -{ 26*z' + 8*z'^2 + 26*z'' + 8*z''^2 }-> 1 + s80 + s81 :|: s80 >= 0, s80 <= z', s81 >= 0, s81 <= z'', z' >= 0, z'' >= 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   times(z', z'') -{ 1 + y + z }-> 1 + s + s' :|: s >= 0, s <= y, s' >= 0, s' <= z, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: 
Previous analysis results are:
  times: runtime: O(n^1) [z''], size: O(n^1) [z'']
  encArg: runtime: O(n^2) [26*z' + 8*z'^2], size: O(n^1) [z']
  encode_+: runtime: O(n^2) [26*z' + 8*z'^2 + 26*z'' + 8*z''^2], size: O(n^1) [1 + z' + z'']
  encode_*: runtime: O(n^2) [3 + 315*z' + 96*z'^2 + 324*z'' + 96*z''^2], size: O(n^1) [z'']

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(45) FinalProof (FINISHED)
Computed overall runtime complexity
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(46)
BOUNDS(1, n^2)

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(47) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(48)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

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

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

   encArg(+(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
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(49) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

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

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [y / +(y, z)].

The result substitution is [ ].




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(50)
Complex Obligation (BEST)

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(51)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

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

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

   encArg(+(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
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(52) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(53)
BOUNDS(n^1, INF)

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(54)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

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

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

   encArg(+(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