WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 41 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
    (7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 133 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 35 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 244 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 466 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 245 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 46 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 147 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 305 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 7 ms]
    (58) CpxRNTS
    (59) FinalProof [FINISHED, 0 ms]
    (60) BOUNDS(1, n^3)
(61) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(62) TRS for Loop Detection
    (63) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (64) BEST
        (65) proven lower bound
            (66) LowerBoundPropagationProof [FINISHED, 0 ms]
            (67) BOUNDS(n^1, INF)
        (68) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

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

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

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

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

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(c_w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   w(x0) -> c_w(x0)

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

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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(c_w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   w(x0) -> c_w(x0)

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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x)) [1]
   b(r(x)) -> r(b(x)) [1]
   b(c_w(x)) -> w(b(x)) [1]
   encArg(r(x_1)) -> r(encArg(x_1)) [0]
   encArg(cons_w(x_1)) -> w(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_w(x_1) -> w(encArg(x_1)) [0]
   encode_r(x_1) -> r(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   w(x0) -> c_w(x0) [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:

   w(r(x)) -> r(w(x)) [1]
   b(r(x)) -> r(b(x)) [1]
   b(c_w(x)) -> w(b(x)) [1]
   encArg(r(x_1)) -> r(encArg(x_1)) [0]
   encArg(cons_w(x_1)) -> w(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_w(x_1) -> w(encArg(x_1)) [0]
   encode_r(x_1) -> r(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   w(x0) -> c_w(x0) [0]

The TRS has the following type information:
w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
r :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
b :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
c_w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encArg :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
cons_w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
cons_b :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encode_w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encode_r :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encode_b :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b

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:

   b_1
   encArg_1
   encode_w_1
   encode_r_1
   encode_b_1
   w_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_w(v0) -> const [0]
   encode_r(v0) -> const [0]
   encode_b(v0) -> const [0]
   w(v0) -> const [0]
   b(v0) -> const [0]

And the following fresh constants:    const

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

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

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

   w(r(x)) -> r(w(x)) [1]
   b(r(x)) -> r(b(x)) [1]
   b(c_w(x)) -> w(b(x)) [1]
   encArg(r(x_1)) -> r(encArg(x_1)) [0]
   encArg(cons_w(x_1)) -> w(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_w(x_1) -> w(encArg(x_1)) [0]
   encode_r(x_1) -> r(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   w(x0) -> c_w(x0) [0]
   encArg(v0) -> const [0]
   encode_w(v0) -> const [0]
   encode_r(v0) -> const [0]
   encode_b(v0) -> const [0]
   w(v0) -> const [0]
   b(v0) -> const [0]

The TRS has the following type information:
w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
r :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
b :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
c_w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encArg :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
cons_w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
cons_b :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encode_w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encode_r :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encode_b :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
const :: r:c_w:cons_w:cons_b:const

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:

   w(r(x)) -> r(w(x)) [1]
   b(r(x)) -> r(b(x)) [1]
   b(c_w(r(x'))) -> w(r(b(x'))) [2]
   b(c_w(c_w(x''))) -> w(w(b(x''))) [2]
   b(c_w(x)) -> w(const) [1]
   encArg(r(x_1)) -> r(encArg(x_1)) [0]
   encArg(cons_w(r(x_1'))) -> w(r(encArg(x_1'))) [0]
   encArg(cons_w(cons_w(x_1''))) -> w(w(encArg(x_1''))) [0]
   encArg(cons_w(cons_b(x_11))) -> w(b(encArg(x_11))) [0]
   encArg(cons_w(x_1)) -> w(const) [0]
   encArg(cons_b(r(x_12))) -> b(r(encArg(x_12))) [0]
   encArg(cons_b(cons_w(x_13))) -> b(w(encArg(x_13))) [0]
   encArg(cons_b(cons_b(x_14))) -> b(b(encArg(x_14))) [0]
   encArg(cons_b(x_1)) -> b(const) [0]
   encode_w(r(x_15)) -> w(r(encArg(x_15))) [0]
   encode_w(cons_w(x_16)) -> w(w(encArg(x_16))) [0]
   encode_w(cons_b(x_17)) -> w(b(encArg(x_17))) [0]
   encode_w(x_1) -> w(const) [0]
   encode_r(x_1) -> r(encArg(x_1)) [0]
   encode_b(r(x_18)) -> b(r(encArg(x_18))) [0]
   encode_b(cons_w(x_19)) -> b(w(encArg(x_19))) [0]
   encode_b(cons_b(x_110)) -> b(b(encArg(x_110))) [0]
   encode_b(x_1) -> b(const) [0]
   w(x0) -> c_w(x0) [0]
   encArg(v0) -> const [0]
   encode_w(v0) -> const [0]
   encode_r(v0) -> const [0]
   encode_b(v0) -> const [0]
   w(v0) -> const [0]
   b(v0) -> const [0]

The TRS has the following type information:
w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
r :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
b :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
c_w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encArg :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
cons_w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
cons_b :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encode_w :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encode_r :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
encode_b :: r:c_w:cons_w:cons_b:const -> r:c_w:cons_w:cons_b:const
const :: r:c_w:cons_w:cons_b:const

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

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

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

   b(z) -{ 2 }-> w(w(b(x''))) :|: x'' >= 0, z = 1 + (1 + x'')
   b(z) -{ 1 }-> w(0) :|: x >= 0, z = 1 + x
   b(z) -{ 2 }-> w(1 + b(x')) :|: x' >= 0, z = 1 + (1 + x')
   b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   b(z) -{ 1 }-> 1 + b(x) :|: x >= 0, z = 1 + x
   encArg(z) -{ 0 }-> w(w(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0
   encArg(z) -{ 0 }-> w(b(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11)
   encArg(z) -{ 0 }-> w(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> b(w(encArg(x_13))) :|: z = 1 + (1 + x_13), x_13 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(x_14))) :|: x_14 >= 0, z = 1 + (1 + x_14)
   encArg(z) -{ 0 }-> b(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(x_12)) :|: z = 1 + (1 + x_12), x_12 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(x_19))) :|: z = 1 + x_19, x_19 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(x_110))) :|: z = 1 + x_110, x_110 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: x_1 >= 0, z = x_1
   encode_b(z) -{ 0 }-> b(1 + encArg(x_18)) :|: z = 1 + x_18, x_18 >= 0
   encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_r(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_r(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_w(z) -{ 0 }-> w(w(encArg(x_16))) :|: z = 1 + x_16, x_16 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(x_17))) :|: x_17 >= 0, z = 1 + x_17
   encode_w(z) -{ 0 }-> w(0) :|: x_1 >= 0, z = x_1
   encode_w(z) -{ 0 }-> w(1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + x_15
   encode_w(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   w(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   w(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   w(z) -{ 1 }-> 1 + w(x) :|: x >= 0, z = 1 + x


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

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

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

   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 1 }-> w(0) :|: z - 1 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(0) :|: z >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0
   w(z) -{ 1 }-> 1 + w(z - 1) :|: z - 1 >= 0


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

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

   { w }
   { b }
   { encArg }
   { encode_b }
   { encode_r }
   { encode_w }

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

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

   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 1 }-> w(0) :|: z - 1 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(0) :|: z >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0
   w(z) -{ 1 }-> 1 + w(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {w}, {b}, {encArg}, {encode_b}, {encode_r}, {encode_w}

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

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

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

   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 1 }-> w(0) :|: z - 1 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(0) :|: z >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0
   w(z) -{ 1 }-> 1 + w(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {w}, {b}, {encArg}, {encode_b}, {encode_r}, {encode_w}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 1 }-> w(0) :|: z - 1 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(0) :|: z >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0
   w(z) -{ 1 }-> 1 + w(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {w}, {b}, {encArg}, {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: ?, size: O(n^1) [1 + z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 1 }-> w(0) :|: z - 1 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(0) :|: z >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0
   w(z) -{ 1 }-> 1 + w(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {b}, {encArg}, {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {b}, {encArg}, {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {b}, {encArg}, {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: ?, size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: b
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 5*z + 2*z^2

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 2 }-> w(w(b(z - 2))) :|: z - 2 >= 0
   b(z) -{ 2 }-> w(1 + b(z - 2)) :|: z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 1 }-> 1 + b(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(0) :|: z >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: ?, size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 4 + 5*z + 6*z^2 + 10*z^3

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> w(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> w(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(w(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(b(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> b(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> b(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> b(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ 0 }-> w(w(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(b(encArg(z - 1))) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> w(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_b}, {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: ?, size: O(n^1) [z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_b
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 13 + 94*z + 30*z^3

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: O(n^3) [13 + 94*z + 30*z^3], size: O(n^1) [z]

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: O(n^3) [13 + 94*z + 30*z^3], size: O(n^1) [z]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_r}, {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: O(n^3) [13 + 94*z + 30*z^3], size: O(n^1) [z]
  encode_r: runtime: ?, size: O(n^1) [1 + z]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_r
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 4 + 5*z + 6*z^2 + 10*z^3

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: O(n^3) [13 + 94*z + 30*z^3], size: O(n^1) [z]
  encode_r: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [1 + z]

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

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

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

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: O(n^3) [13 + 94*z + 30*z^3], size: O(n^1) [z]
  encode_r: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [1 + z]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_w}
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: O(n^3) [13 + 94*z + 30*z^3], size: O(n^1) [z]
  encode_r: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [1 + z]
  encode_w: runtime: ?, size: O(n^1) [1 + z]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_w
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 78*z + 30*z^3

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   b(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0 + 1, z - 1 >= 0
   b(z) -{ 5 + s3 + -3*z + 2*z^2 }-> s4 :|: s3 >= 0, s3 <= z - 2, s4 >= 0, s4 <= 1 + s3 + 1, z - 2 >= 0
   b(z) -{ 4 + s5 + s6 + -3*z + 2*z^2 }-> s7 :|: s5 >= 0, s5 <= z - 2, s6 >= 0, s6 <= s5 + 1, s7 >= 0, s7 <= s6 + 1, z - 2 >= 0
   b(z) -{ 0 }-> 0 :|: z >= 0
   b(z) -{ 2 + z + 2*z^2 }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z - 1 >= 0
   encArg(z) -{ -61 + s11 + 101*z + -54*z^2 + 10*z^3 }-> s12 :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + s11 + 1, z - 2 >= 0
   encArg(z) -{ -62 + s13 + s14 + 101*z + -54*z^2 + 10*z^3 }-> s15 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= s13 + 1, s15 >= 0, s15 <= s14 + 1, z - 2 >= 0
   encArg(z) -{ -58 + 5*s16 + 2*s16^2 + s17 + 101*z + -54*z^2 + 10*z^3 }-> s18 :|: s16 >= 0, s16 <= z - 2, s17 >= 0, s17 <= s16, s18 >= 0, s18 <= s17 + 1, z - 2 >= 0
   encArg(z) -{ -51 + 9*s19 + 2*s19^2 + 101*z + -54*z^2 + 10*z^3 }-> s20 :|: s19 >= 0, s19 <= z - 2, s20 >= 0, s20 <= 1 + s19, z - 2 >= 0
   encArg(z) -{ -58 + s21 + 5*s22 + 2*s22^2 + 101*z + -54*z^2 + 10*z^3 }-> s23 :|: s21 >= 0, s21 <= z - 2, s22 >= 0, s22 <= s21 + 1, s23 >= 0, s23 <= s22, z - 2 >= 0
   encArg(z) -{ -54 + 5*s24 + 2*s24^2 + 5*s25 + 2*s25^2 + 101*z + -54*z^2 + 10*z^3 }-> s26 :|: s24 >= 0, s24 <= z - 2, s25 >= 0, s25 <= s24, s26 >= 0, s26 <= s25, z - 2 >= 0
   encArg(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -5 + 23*z + -24*z^2 + 10*z^3 }-> 1 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0
   encode_b(z) -{ 6 + 9*s36 + 2*s36^2 + 23*z + -24*z^2 + 10*z^3 }-> s37 :|: s36 >= 0, s36 <= z - 1, s37 >= 0, s37 <= 1 + s36, z - 1 >= 0
   encode_b(z) -{ -1 + s38 + 5*s39 + 2*s39^2 + 23*z + -24*z^2 + 10*z^3 }-> s40 :|: s38 >= 0, s38 <= z - 1, s39 >= 0, s39 <= s38 + 1, s40 >= 0, s40 <= s39, z - 1 >= 0
   encode_b(z) -{ 3 + 5*s41 + 2*s41^2 + 5*s42 + 2*s42^2 + 23*z + -24*z^2 + 10*z^3 }-> s43 :|: s41 >= 0, s41 <= z - 1, s42 >= 0, s42 <= s41, s43 >= 0, s43 <= s42, z - 1 >= 0
   encode_b(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0, z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 0 }-> 0 :|: z >= 0
   encode_r(z) -{ 4 + 5*z + 6*z^2 + 10*z^3 }-> 1 + s35 :|: s35 >= 0, s35 <= z, z >= 0
   encode_w(z) -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z >= 0
   encode_w(z) -{ -4 + s27 + 23*z + -24*z^2 + 10*z^3 }-> s28 :|: s27 >= 0, s27 <= z - 1, s28 >= 0, s28 <= 1 + s27 + 1, z - 1 >= 0
   encode_w(z) -{ -5 + s29 + s30 + 23*z + -24*z^2 + 10*z^3 }-> s31 :|: s29 >= 0, s29 <= z - 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z - 1 >= 0
   encode_w(z) -{ -1 + 5*s32 + 2*s32^2 + s33 + 23*z + -24*z^2 + 10*z^3 }-> s34 :|: s32 >= 0, s32 <= z - 1, s33 >= 0, s33 <= s32, s34 >= 0, s34 <= s33 + 1, z - 1 >= 0
   encode_w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ 0 }-> 0 :|: z >= 0
   w(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 1 + 1, z - 1 >= 0
   w(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  w: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  b: runtime: O(n^2) [4 + 5*z + 2*z^2], size: O(n^1) [z]
  encArg: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [z]
  encode_b: runtime: O(n^3) [13 + 94*z + 30*z^3], size: O(n^1) [z]
  encode_r: runtime: O(n^3) [4 + 5*z + 6*z^2 + 10*z^3], size: O(n^1) [1 + z]
  encode_w: runtime: O(n^3) [78*z + 30*z^3], size: O(n^1) [1 + z]

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

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

(60)
BOUNDS(1, n^3)

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

(61) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(62)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

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

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

The rewrite sequence

b(r(x)) ->^+ r(b(x))

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

The pumping substitution is [x / r(x)].

The result substitution is [ ].




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

(64)
Complex Obligation (BEST)

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

(65)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

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

(66) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(67)
BOUNDS(n^1, INF)

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

(68)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

Rewrite Strategy: FULL