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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 164 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) SimplificationProof [BOTH BOUNDS(ID, ID), 2 ms]
    (20) CpxRNTS
    (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 211 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 982 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 263 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2035 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 877 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 266 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 1238 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 763 ms]
    (58) CpxRNTS
    (59) FinalProof [FINISHED, 0 ms]
    (60) BOUNDS(1, n^3)
(61) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(62) CpxRelTRS
    (63) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) typed CpxTrs
    (65) OrderProof [LOWER BOUND(ID), 0 ms]
    (66) typed CpxTrs
    (67) RewriteLemmaProof [LOWER BOUND(ID), 240 ms]
    (68) proven lower bound
    (69) LowerBoundPropagationProof [FINISHED, 0 ms]
    (70) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(0, s(y)) -> s(y)
   s(+(0, y)) -> s(y)

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(0) -> 0
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))


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

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


The TRS R consists of the following rules:

   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(0, s(y)) -> s(y)
   s(+(0, y)) -> s(y)

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

   encArg(0) -> 0
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(0, s(y)) -> s(y)
   s(+(0, y)) -> s(y)

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

   encArg(0) -> 0
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   +(x, 0) -> x
   +(0, c_s(y)) -> s(y)
   s(c_+(0, y)) -> s(y)
   +(x, c_s(y)) -> s(+(x, y))

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

   encArg(0) -> 0
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   s(x0) -> c_s(x0)
   +(x0, x1) -> c_+(x0, x1)

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

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

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


The TRS R consists of the following rules:

   +(x, 0) -> x [1]
   +(0, c_s(y)) -> s(y) [1]
   s(c_+(0, y)) -> s(y) [1]
   +(x, c_s(y)) -> s(+(x, y)) [1]
   encArg(0) -> 0 [0]
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_s(x_1)) -> s(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   s(x0) -> c_s(x0) [0]
   +(x0, x1) -> c_+(x0, x1) [0]

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

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

+ => plus

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

(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:

   plus(x, 0) -> x [1]
   plus(0, c_s(y)) -> s(y) [1]
   s(c_+(0, y)) -> s(y) [1]
   plus(x, c_s(y)) -> s(plus(x, y)) [1]
   encArg(0) -> 0 [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_s(x_1)) -> s(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   s(x0) -> c_s(x0) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]

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

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

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

   plus(x, 0) -> x [1]
   plus(0, c_s(y)) -> s(y) [1]
   s(c_+(0, y)) -> s(y) [1]
   plus(x, c_s(y)) -> s(plus(x, y)) [1]
   encArg(0) -> 0 [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_s(x_1)) -> s(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   s(x0) -> c_s(x0) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]

The TRS has the following type information:
plus :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
0 :: 0:c_s:c_+:cons_+:cons_s
c_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
c_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encArg :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
cons_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
cons_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encode_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encode_0 :: 0:c_s:c_+:cons_+:cons_s
encode_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s

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

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

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

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

(c) The following functions are completely defined:

   encArg_1
   encode_+_2
   encode_0
   encode_s_1
   s_1
   plus_2

Due to the following rules being added:

   encArg(v0) -> 0 [0]
   encode_+(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   s(v0) -> 0 [0]
   plus(v0, v1) -> 0 [0]

And the following fresh constants: none

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

(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:

   plus(x, 0) -> x [1]
   plus(0, c_s(y)) -> s(y) [1]
   s(c_+(0, y)) -> s(y) [1]
   plus(x, c_s(y)) -> s(plus(x, y)) [1]
   encArg(0) -> 0 [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_s(x_1)) -> s(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   s(x0) -> c_s(x0) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   encArg(v0) -> 0 [0]
   encode_+(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   s(v0) -> 0 [0]
   plus(v0, v1) -> 0 [0]

The TRS has the following type information:
plus :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
0 :: 0:c_s:c_+:cons_+:cons_s
c_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
c_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encArg :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
cons_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
cons_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encode_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encode_0 :: 0:c_s:c_+:cons_+:cons_s
encode_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s

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:

   plus(x, 0) -> x [1]
   plus(0, c_s(y)) -> s(y) [1]
   s(c_+(0, y)) -> s(y) [1]
   plus(x, c_s(0)) -> s(x) [2]
   plus(0, c_s(c_s(y'))) -> s(s(y')) [2]
   plus(x, c_s(c_s(y''))) -> s(s(plus(x, y''))) [2]
   plus(x, c_s(y)) -> s(c_+(x, y)) [1]
   plus(x, c_s(y)) -> s(0) [1]
   encArg(0) -> 0 [0]
   encArg(cons_+(0, 0)) -> plus(0, 0) [0]
   encArg(cons_+(0, cons_+(x_11, x_2''))) -> plus(0, plus(encArg(x_11), encArg(x_2''))) [0]
   encArg(cons_+(0, cons_s(x_12))) -> plus(0, s(encArg(x_12))) [0]
   encArg(cons_+(0, x_2)) -> plus(0, 0) [0]
   encArg(cons_+(cons_+(x_1', x_2'), 0)) -> plus(plus(encArg(x_1'), encArg(x_2')), 0) [0]
   encArg(cons_+(cons_+(x_1', x_2'), cons_+(x_13, x_21))) -> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) [0]
   encArg(cons_+(cons_+(x_1', x_2'), cons_s(x_14))) -> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) [0]
   encArg(cons_+(cons_+(x_1', x_2'), x_2)) -> plus(plus(encArg(x_1'), encArg(x_2')), 0) [0]
   encArg(cons_+(cons_s(x_1''), 0)) -> plus(s(encArg(x_1'')), 0) [0]
   encArg(cons_+(cons_s(x_1''), cons_+(x_15, x_22))) -> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) [0]
   encArg(cons_+(cons_s(x_1''), cons_s(x_16))) -> plus(s(encArg(x_1'')), s(encArg(x_16))) [0]
   encArg(cons_+(cons_s(x_1''), x_2)) -> plus(s(encArg(x_1'')), 0) [0]
   encArg(cons_+(x_1, 0)) -> plus(0, 0) [0]
   encArg(cons_+(x_1, cons_+(x_17, x_23))) -> plus(0, plus(encArg(x_17), encArg(x_23))) [0]
   encArg(cons_+(x_1, cons_s(x_18))) -> plus(0, s(encArg(x_18))) [0]
   encArg(cons_+(x_1, x_2)) -> plus(0, 0) [0]
   encArg(cons_s(0)) -> s(0) [0]
   encArg(cons_s(cons_+(x_19, x_24))) -> s(plus(encArg(x_19), encArg(x_24))) [0]
   encArg(cons_s(cons_s(x_110))) -> s(s(encArg(x_110))) [0]
   encArg(cons_s(x_1)) -> s(0) [0]
   encode_+(0, 0) -> plus(0, 0) [0]
   encode_+(0, cons_+(x_113, x_26)) -> plus(0, plus(encArg(x_113), encArg(x_26))) [0]
   encode_+(0, cons_s(x_114)) -> plus(0, s(encArg(x_114))) [0]
   encode_+(0, x_2) -> plus(0, 0) [0]
   encode_+(cons_+(x_111, x_25), 0) -> plus(plus(encArg(x_111), encArg(x_25)), 0) [0]
   encode_+(cons_+(x_111, x_25), cons_+(x_115, x_27)) -> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) [0]
   encode_+(cons_+(x_111, x_25), cons_s(x_116)) -> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(x_116))) [0]
   encode_+(cons_+(x_111, x_25), x_2) -> plus(plus(encArg(x_111), encArg(x_25)), 0) [0]
   encode_+(cons_s(x_112), 0) -> plus(s(encArg(x_112)), 0) [0]
   encode_+(cons_s(x_112), cons_+(x_117, x_28)) -> plus(s(encArg(x_112)), plus(encArg(x_117), encArg(x_28))) [0]
   encode_+(cons_s(x_112), cons_s(x_118)) -> plus(s(encArg(x_112)), s(encArg(x_118))) [0]
   encode_+(cons_s(x_112), x_2) -> plus(s(encArg(x_112)), 0) [0]
   encode_+(x_1, 0) -> plus(0, 0) [0]
   encode_+(x_1, cons_+(x_119, x_29)) -> plus(0, plus(encArg(x_119), encArg(x_29))) [0]
   encode_+(x_1, cons_s(x_120)) -> plus(0, s(encArg(x_120))) [0]
   encode_+(x_1, x_2) -> plus(0, 0) [0]
   encode_0 -> 0 [0]
   encode_s(0) -> s(0) [0]
   encode_s(cons_+(x_121, x_210)) -> s(plus(encArg(x_121), encArg(x_210))) [0]
   encode_s(cons_s(x_122)) -> s(s(encArg(x_122))) [0]
   encode_s(x_1) -> s(0) [0]
   s(x0) -> c_s(x0) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   encArg(v0) -> 0 [0]
   encode_+(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   s(v0) -> 0 [0]
   plus(v0, v1) -> 0 [0]

The TRS has the following type information:
plus :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
0 :: 0:c_s:c_+:cons_+:cons_s
c_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
c_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encArg :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
cons_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
cons_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encode_+ :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s
encode_0 :: 0:c_s:c_+:cons_+:cons_s
encode_s :: 0:c_s:c_+:cons_+:cons_s -> 0:c_s:c_+:cons_+:cons_s

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:

   0 => 0

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

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

   encArg(z) -{ 0 }-> s(s(encArg(x_110))) :|: z = 1 + (1 + x_110), x_110 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_12))) :|: z = 1 + 0 + (1 + x_12), x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + x_1 + 0, x_1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(x_112)), s(encArg(x_118))) :|: z = 1 + x_112, z' = 1 + x_118, x_112 >= 0, x_118 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(x_112)), plus(encArg(x_117), encArg(x_28))) :|: z = 1 + x_112, x_117 >= 0, z' = 1 + x_117 + x_28, x_112 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(x_112)), 0) :|: z = 1 + x_112, x_112 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(x_112)), 0) :|: z = 1 + x_112, x_2 >= 0, z' = x_2, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(x_116))) :|: x_116 >= 0, x_25 >= 0, z' = 1 + x_116, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, x_2 >= 0, z' = x_2, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(x_114))) :|: x_114 >= 0, z = 0, z' = 1 + x_114
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(x_120))) :|: x_1 >= 0, x_120 >= 0, z = x_1, z' = 1 + x_120
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: x_1 >= 0, z = x_1, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: x_2 >= 0, z' = x_2, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = x_1, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_+(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(x_122))) :|: x_122 >= 0, z = 1 + x_122
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: x_1 >= 0, z = x_1
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 2 }-> s(x) :|: x >= 0, z' = 1 + 0, z = x
   plus(z, z') -{ 1 }-> s(y) :|: z' = 1 + y, y >= 0, z = 0
   plus(z, z') -{ 2 }-> s(s(y')) :|: z' = 1 + (1 + y'), y' >= 0, z = 0
   plus(z, z') -{ 2 }-> s(s(plus(x, y''))) :|: z' = 1 + (1 + y''), x >= 0, y'' >= 0, z = x
   plus(z, z') -{ 1 }-> s(0) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   plus(z, z') -{ 1 }-> s(1 + x + y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1
   s(z) -{ 1 }-> s(y) :|: y >= 0, z = 1 + 0 + y
   s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   s(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0


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

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0


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

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

   { encode_0 }
   { s }
   { plus }
   { encArg }
   { encode_s }
   { encode_+ }

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_0}, {s}, {plus}, {encArg}, {encode_s}, {encode_+}

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

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_0}, {s}, {plus}, {encArg}, {encode_s}, {encode_+}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_0}, {s}, {plus}, {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: ?, size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {s}, {plus}, {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {s}, {plus}, {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {s}, {plus}, {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: ?, size: O(n^1) [1 + z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> s(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> s(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> s(0) :|: z = 0
   encode_s(z) -{ 0 }-> s(0) :|: z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(z) :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 2 }-> s(s(z' - 2)) :|: z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(0) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s(z' - 1) :|: z' - 1 >= 0, z = 0
   plus(z, z') -{ 1 }-> s(1 + z + (z' - 1)) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ 1 }-> s(z - 1) :|: z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {plus}, {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [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:

   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {plus}, {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {plus}, {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: ?, size: O(n^1) [1 + z + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> plus(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> plus(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z = 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z' >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(0, 0) :|: z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 2 }-> s(s(plus(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + 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:

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: ?, size: O(n^1) [1 + z]

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

(45) 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: 498 + 1407*z + 916*z^2 + 188*z^3

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

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

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 0 }-> s(s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> s(plus(encArg(x_19), encArg(x_24))) :|: z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), s(encArg(x_16))) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), plus(encArg(x_15), encArg(x_22))) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(x_1'')), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(s(encArg(z - 2)), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), s(encArg(x_14))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), plus(encArg(x_13), encArg(x_21))) :|: z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(x_18))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, s(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_11), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_17), encArg(x_23))) :|: x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), s(encArg(z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), plus(encArg(x_117), encArg(x_28))) :|: x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(s(encArg(z - 1)), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), s(encArg(z' - 1))) :|: z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), plus(encArg(x_115), encArg(x_27))) :|: x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_111), encArg(x_25)), 0) :|: x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, s(encArg(z' - 1))) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_113), encArg(x_26))) :|: x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_119), encArg(x_29))) :|: z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> s(s(encArg(z - 1))) :|: z - 1 >= 0
   encode_s(z) -{ 0 }-> s(plus(encArg(x_121), encArg(x_210))) :|: z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^3) [498 + 1407*z + 916*z^2 + 188*z^3], size: O(n^1) [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:

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s19 + 2*s19*s20 + 5*s20 + 2*s20^2 + 5*s21 + 2*s21^2 + 1407*x_11 + 916*x_11^2 + 188*x_11^3 + 1407*x_2'' + 916*x_2''^2 + 188*x_2''^3 }-> s22 :|: s19 >= 0, s19 <= x_11 + 1, s20 >= 0, s20 <= x_2'' + 1, s21 >= 0, s21 <= s19 + s20 + 1, s22 >= 0, s22 <= 0 + s21 + 1, z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ -151 + s23 + 5*s24 + 2*s24^2 + -1*z + -212*z^2 + 188*z^3 }-> s25 :|: s23 >= 0, s23 <= z - 2 + 1, s24 >= 0, s24 <= s23 + 1, s25 >= 0, s25 <= 0 + s24 + 1, z - 2 >= 0
   encArg(z) -{ 1006 + 2*s26 + 2*s26*s27 + 5*s27 + 2*s27^2 + 2*s28 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s29 :|: s26 >= 0, s26 <= x_1' + 1, s27 >= 0, s27 <= x_2' + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s28 + 0 + 1, x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 2007 + 2*s30 + 2*s30*s31 + 5*s31 + 2*s31^2 + 2*s32 + 2*s32*s35 + 2*s33 + 2*s33*s34 + 5*s34 + 2*s34^2 + 5*s35 + 2*s35^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_13 + 916*x_13^2 + 188*x_13^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 + 1407*x_21 + 916*x_21^2 + 188*x_21^3 }-> s36 :|: s30 >= 0, s30 <= x_1' + 1, s31 >= 0, s31 <= x_2' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= x_13 + 1, s34 >= 0, s34 <= x_21 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s32 + s35 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 1504 + 2*s37 + 2*s37*s38 + 5*s38 + 2*s38^2 + 2*s39 + 2*s39*s41 + s40 + 5*s41 + 2*s41^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_14 + 916*x_14^2 + 188*x_14^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s42 :|: s37 >= 0, s37 <= x_1' + 1, s38 >= 0, s38 <= x_2' + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= x_14 + 1, s41 >= 0, s41 <= s40 + 1, s42 >= 0, s42 <= s39 + s41 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 1006 + 2*s43 + 2*s43*s44 + 5*s44 + 2*s44^2 + 2*s45 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s46 :|: s43 >= 0, s43 <= x_1' + 1, s44 >= 0, s44 <= x_2' + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s45 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ -151 + s47 + 2*s48 + -1*z + -212*z^2 + 188*z^3 }-> s49 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= s47 + 1, s49 >= 0, s49 <= s48 + 0 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 1504 + s50 + 2*s51 + 2*s51*s54 + 2*s52 + 2*s52*s53 + 5*s53 + 2*s53^2 + 5*s54 + 2*s54^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_15 + 916*x_15^2 + 188*x_15^3 + 1407*x_22 + 916*x_22^2 + 188*x_22^3 }-> s55 :|: s50 >= 0, s50 <= x_1'' + 1, s51 >= 0, s51 <= s50 + 1, s52 >= 0, s52 <= x_15 + 1, s53 >= 0, s53 <= x_22 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s51 + s54 + 1, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1001 + s56 + 2*s57 + 2*s57*s59 + s58 + 5*s59 + 2*s59^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_16 + 916*x_16^2 + 188*x_16^3 }-> s60 :|: s56 >= 0, s56 <= x_1'' + 1, s57 >= 0, s57 <= s56 + 1, s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s57 + s59 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 503 + s61 + 2*s62 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 }-> s63 :|: s61 >= 0, s61 <= x_1'' + 1, s62 >= 0, s62 <= s61 + 1, s63 >= 0, s63 <= s62 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s64 + 2*s64*s65 + 5*s65 + 2*s65^2 + 5*s66 + 2*s66^2 + 1407*x_17 + 916*x_17^2 + 188*x_17^3 + 1407*x_23 + 916*x_23^2 + 188*x_23^3 }-> s67 :|: s64 >= 0, s64 <= x_17 + 1, s65 >= 0, s65 <= x_23 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= 0 + s66 + 1, x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 503 + s68 + 5*s69 + 2*s69^2 + 1407*x_18 + 916*x_18^2 + 188*x_18^3 }-> s70 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= s68 + 1, s70 >= 0, s70 <= 0 + s69 + 1, x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 1001 + 2*s71 + 2*s71*s72 + 5*s72 + 2*s72^2 + s73 + 1407*x_19 + 916*x_19^2 + 188*x_19^3 + 1407*x_24 + 916*x_24^2 + 188*x_24^3 }-> s74 :|: s71 >= 0, s71 <= x_19 + 1, s72 >= 0, s72 <= x_24 + 1, s73 >= 0, s73 <= s71 + s72 + 1, s74 >= 0, s74 <= s73 + 1, z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ -156 + s75 + s76 + -1*z + -212*z^2 + 188*z^3 }-> s77 :|: s75 >= 0, s75 <= z - 2 + 1, s76 >= 0, s76 <= s75 + 1, s77 >= 0, s77 <= s76 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 825 + 5*s100 + 2*s100*s98 + 2*s100^2 + 2*s96 + 2*s96*s97 + 5*s97 + 2*s97^2 + 2*s98 + s99 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s101 :|: s96 >= 0, s96 <= x_111 + 1, s97 >= 0, s97 <= x_25 + 1, s98 >= 0, s98 <= s96 + s97 + 1, s99 >= 0, s99 <= z' - 1 + 1, s100 >= 0, s100 <= s99 + 1, s101 >= 0, s101 <= s98 + s100 + 1, z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 1006 + 2*s102 + 2*s102*s103 + 5*s103 + 2*s103^2 + 2*s104 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s105 :|: s102 >= 0, s102 <= x_111 + 1, s103 >= 0, s103 <= x_25 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= s104 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ -176 + s106 + 2*s107 + 139*z + 352*z^2 + 188*z^3 }-> s108 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= s106 + 1, s108 >= 0, s108 <= s107 + 0 + 1, z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 825 + s109 + 2*s110 + 2*s110*s113 + 2*s111 + 2*s111*s112 + 5*s112 + 2*s112^2 + 5*s113 + 2*s113^2 + 1407*x_117 + 916*x_117^2 + 188*x_117^3 + 1407*x_28 + 916*x_28^2 + 188*x_28^3 + 139*z + 352*z^2 + 188*z^3 }-> s114 :|: s109 >= 0, s109 <= z - 1 + 1, s110 >= 0, s110 <= s109 + 1, s111 >= 0, s111 <= x_117 + 1, s112 >= 0, s112 <= x_28 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s110 + s113 + 1, x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ -357 + s115 + 2*s116 + 2*s116*s118 + s117 + 5*s118 + 2*s118^2 + 139*z + 352*z^2 + 188*z^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s119 :|: s115 >= 0, s115 <= z - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= z' - 1 + 1, s118 >= 0, s118 <= s117 + 1, s119 >= 0, s119 <= s116 + s118 + 1, z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ -176 + s120 + 2*s121 + 139*z + 352*z^2 + 188*z^3 }-> s122 :|: s120 >= 0, s120 <= z - 1 + 1, s121 >= 0, s121 <= s120 + 1, s122 >= 0, s122 <= s121 + 0 + 1, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 1006 + 2*s123 + 2*s123*s124 + 5*s124 + 2*s124^2 + 5*s125 + 2*s125^2 + 1407*x_119 + 916*x_119^2 + 188*x_119^3 + 1407*x_29 + 916*x_29^2 + 188*x_29^3 }-> s126 :|: s123 >= 0, s123 <= x_119 + 1, s124 >= 0, s124 <= x_29 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= 0 + s125 + 1, z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ -176 + s127 + 5*s128 + 2*s128^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s129 :|: s127 >= 0, s127 <= z' - 1 + 1, s128 >= 0, s128 <= s127 + 1, s129 >= 0, s129 <= 0 + s128 + 1, z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 1006 + 2*s78 + 2*s78*s79 + 5*s79 + 2*s79^2 + 5*s80 + 2*s80^2 + 1407*x_113 + 916*x_113^2 + 188*x_113^3 + 1407*x_26 + 916*x_26^2 + 188*x_26^3 }-> s81 :|: s78 >= 0, s78 <= x_113 + 1, s79 >= 0, s79 <= x_26 + 1, s80 >= 0, s80 <= s78 + s79 + 1, s81 >= 0, s81 <= 0 + s80 + 1, x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ -176 + s82 + 5*s83 + 2*s83^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s84 :|: s82 >= 0, s82 <= z' - 1 + 1, s83 >= 0, s83 <= s82 + 1, s84 >= 0, s84 <= 0 + s83 + 1, z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 1006 + 2*s85 + 2*s85*s86 + 5*s86 + 2*s86^2 + 2*s87 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s88 :|: s85 >= 0, s85 <= x_111 + 1, s86 >= 0, s86 <= x_25 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 2007 + 2*s89 + 2*s89*s90 + 5*s90 + 2*s90^2 + 2*s91 + 2*s91*s94 + 2*s92 + 2*s92*s93 + 5*s93 + 2*s93^2 + 5*s94 + 2*s94^2 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_115 + 916*x_115^2 + 188*x_115^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 1407*x_27 + 916*x_27^2 + 188*x_27^3 }-> s95 :|: s89 >= 0, s89 <= x_111 + 1, s90 >= 0, s90 <= x_25 + 1, s91 >= 0, s91 <= s89 + s90 + 1, s92 >= 0, s92 <= x_115 + 1, s93 >= 0, s93 <= x_27 + 1, s94 >= 0, s94 <= s92 + s93 + 1, s95 >= 0, s95 <= s91 + s94 + 1, x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 1001 + 2*s130 + 2*s130*s131 + 5*s131 + 2*s131^2 + s132 + 1407*x_121 + 916*x_121^2 + 188*x_121^3 + 1407*x_210 + 916*x_210^2 + 188*x_210^3 }-> s133 :|: s130 >= 0, s130 <= x_121 + 1, s131 >= 0, s131 <= x_210 + 1, s132 >= 0, s132 <= s130 + s131 + 1, s133 >= 0, s133 <= s132 + 1, z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ -181 + s134 + s135 + 139*z + 352*z^2 + 188*z^3 }-> s136 :|: s134 >= 0, s134 <= z - 1 + 1, s135 >= 0, s135 <= s134 + 1, s136 >= 0, s136 <= s135 + 1, z - 1 >= 0
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^3) [498 + 1407*z + 916*z^2 + 188*z^3], size: O(n^1) [1 + z]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z

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

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

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s19 + 2*s19*s20 + 5*s20 + 2*s20^2 + 5*s21 + 2*s21^2 + 1407*x_11 + 916*x_11^2 + 188*x_11^3 + 1407*x_2'' + 916*x_2''^2 + 188*x_2''^3 }-> s22 :|: s19 >= 0, s19 <= x_11 + 1, s20 >= 0, s20 <= x_2'' + 1, s21 >= 0, s21 <= s19 + s20 + 1, s22 >= 0, s22 <= 0 + s21 + 1, z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ -151 + s23 + 5*s24 + 2*s24^2 + -1*z + -212*z^2 + 188*z^3 }-> s25 :|: s23 >= 0, s23 <= z - 2 + 1, s24 >= 0, s24 <= s23 + 1, s25 >= 0, s25 <= 0 + s24 + 1, z - 2 >= 0
   encArg(z) -{ 1006 + 2*s26 + 2*s26*s27 + 5*s27 + 2*s27^2 + 2*s28 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s29 :|: s26 >= 0, s26 <= x_1' + 1, s27 >= 0, s27 <= x_2' + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s28 + 0 + 1, x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 2007 + 2*s30 + 2*s30*s31 + 5*s31 + 2*s31^2 + 2*s32 + 2*s32*s35 + 2*s33 + 2*s33*s34 + 5*s34 + 2*s34^2 + 5*s35 + 2*s35^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_13 + 916*x_13^2 + 188*x_13^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 + 1407*x_21 + 916*x_21^2 + 188*x_21^3 }-> s36 :|: s30 >= 0, s30 <= x_1' + 1, s31 >= 0, s31 <= x_2' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= x_13 + 1, s34 >= 0, s34 <= x_21 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s32 + s35 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 1504 + 2*s37 + 2*s37*s38 + 5*s38 + 2*s38^2 + 2*s39 + 2*s39*s41 + s40 + 5*s41 + 2*s41^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_14 + 916*x_14^2 + 188*x_14^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s42 :|: s37 >= 0, s37 <= x_1' + 1, s38 >= 0, s38 <= x_2' + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= x_14 + 1, s41 >= 0, s41 <= s40 + 1, s42 >= 0, s42 <= s39 + s41 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 1006 + 2*s43 + 2*s43*s44 + 5*s44 + 2*s44^2 + 2*s45 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s46 :|: s43 >= 0, s43 <= x_1' + 1, s44 >= 0, s44 <= x_2' + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s45 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ -151 + s47 + 2*s48 + -1*z + -212*z^2 + 188*z^3 }-> s49 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= s47 + 1, s49 >= 0, s49 <= s48 + 0 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 1504 + s50 + 2*s51 + 2*s51*s54 + 2*s52 + 2*s52*s53 + 5*s53 + 2*s53^2 + 5*s54 + 2*s54^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_15 + 916*x_15^2 + 188*x_15^3 + 1407*x_22 + 916*x_22^2 + 188*x_22^3 }-> s55 :|: s50 >= 0, s50 <= x_1'' + 1, s51 >= 0, s51 <= s50 + 1, s52 >= 0, s52 <= x_15 + 1, s53 >= 0, s53 <= x_22 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s51 + s54 + 1, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1001 + s56 + 2*s57 + 2*s57*s59 + s58 + 5*s59 + 2*s59^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_16 + 916*x_16^2 + 188*x_16^3 }-> s60 :|: s56 >= 0, s56 <= x_1'' + 1, s57 >= 0, s57 <= s56 + 1, s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s57 + s59 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 503 + s61 + 2*s62 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 }-> s63 :|: s61 >= 0, s61 <= x_1'' + 1, s62 >= 0, s62 <= s61 + 1, s63 >= 0, s63 <= s62 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s64 + 2*s64*s65 + 5*s65 + 2*s65^2 + 5*s66 + 2*s66^2 + 1407*x_17 + 916*x_17^2 + 188*x_17^3 + 1407*x_23 + 916*x_23^2 + 188*x_23^3 }-> s67 :|: s64 >= 0, s64 <= x_17 + 1, s65 >= 0, s65 <= x_23 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= 0 + s66 + 1, x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 503 + s68 + 5*s69 + 2*s69^2 + 1407*x_18 + 916*x_18^2 + 188*x_18^3 }-> s70 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= s68 + 1, s70 >= 0, s70 <= 0 + s69 + 1, x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 1001 + 2*s71 + 2*s71*s72 + 5*s72 + 2*s72^2 + s73 + 1407*x_19 + 916*x_19^2 + 188*x_19^3 + 1407*x_24 + 916*x_24^2 + 188*x_24^3 }-> s74 :|: s71 >= 0, s71 <= x_19 + 1, s72 >= 0, s72 <= x_24 + 1, s73 >= 0, s73 <= s71 + s72 + 1, s74 >= 0, s74 <= s73 + 1, z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ -156 + s75 + s76 + -1*z + -212*z^2 + 188*z^3 }-> s77 :|: s75 >= 0, s75 <= z - 2 + 1, s76 >= 0, s76 <= s75 + 1, s77 >= 0, s77 <= s76 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 825 + 5*s100 + 2*s100*s98 + 2*s100^2 + 2*s96 + 2*s96*s97 + 5*s97 + 2*s97^2 + 2*s98 + s99 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s101 :|: s96 >= 0, s96 <= x_111 + 1, s97 >= 0, s97 <= x_25 + 1, s98 >= 0, s98 <= s96 + s97 + 1, s99 >= 0, s99 <= z' - 1 + 1, s100 >= 0, s100 <= s99 + 1, s101 >= 0, s101 <= s98 + s100 + 1, z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 1006 + 2*s102 + 2*s102*s103 + 5*s103 + 2*s103^2 + 2*s104 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s105 :|: s102 >= 0, s102 <= x_111 + 1, s103 >= 0, s103 <= x_25 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= s104 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ -176 + s106 + 2*s107 + 139*z + 352*z^2 + 188*z^3 }-> s108 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= s106 + 1, s108 >= 0, s108 <= s107 + 0 + 1, z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 825 + s109 + 2*s110 + 2*s110*s113 + 2*s111 + 2*s111*s112 + 5*s112 + 2*s112^2 + 5*s113 + 2*s113^2 + 1407*x_117 + 916*x_117^2 + 188*x_117^3 + 1407*x_28 + 916*x_28^2 + 188*x_28^3 + 139*z + 352*z^2 + 188*z^3 }-> s114 :|: s109 >= 0, s109 <= z - 1 + 1, s110 >= 0, s110 <= s109 + 1, s111 >= 0, s111 <= x_117 + 1, s112 >= 0, s112 <= x_28 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s110 + s113 + 1, x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ -357 + s115 + 2*s116 + 2*s116*s118 + s117 + 5*s118 + 2*s118^2 + 139*z + 352*z^2 + 188*z^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s119 :|: s115 >= 0, s115 <= z - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= z' - 1 + 1, s118 >= 0, s118 <= s117 + 1, s119 >= 0, s119 <= s116 + s118 + 1, z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ -176 + s120 + 2*s121 + 139*z + 352*z^2 + 188*z^3 }-> s122 :|: s120 >= 0, s120 <= z - 1 + 1, s121 >= 0, s121 <= s120 + 1, s122 >= 0, s122 <= s121 + 0 + 1, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 1006 + 2*s123 + 2*s123*s124 + 5*s124 + 2*s124^2 + 5*s125 + 2*s125^2 + 1407*x_119 + 916*x_119^2 + 188*x_119^3 + 1407*x_29 + 916*x_29^2 + 188*x_29^3 }-> s126 :|: s123 >= 0, s123 <= x_119 + 1, s124 >= 0, s124 <= x_29 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= 0 + s125 + 1, z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ -176 + s127 + 5*s128 + 2*s128^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s129 :|: s127 >= 0, s127 <= z' - 1 + 1, s128 >= 0, s128 <= s127 + 1, s129 >= 0, s129 <= 0 + s128 + 1, z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 1006 + 2*s78 + 2*s78*s79 + 5*s79 + 2*s79^2 + 5*s80 + 2*s80^2 + 1407*x_113 + 916*x_113^2 + 188*x_113^3 + 1407*x_26 + 916*x_26^2 + 188*x_26^3 }-> s81 :|: s78 >= 0, s78 <= x_113 + 1, s79 >= 0, s79 <= x_26 + 1, s80 >= 0, s80 <= s78 + s79 + 1, s81 >= 0, s81 <= 0 + s80 + 1, x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ -176 + s82 + 5*s83 + 2*s83^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s84 :|: s82 >= 0, s82 <= z' - 1 + 1, s83 >= 0, s83 <= s82 + 1, s84 >= 0, s84 <= 0 + s83 + 1, z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 1006 + 2*s85 + 2*s85*s86 + 5*s86 + 2*s86^2 + 2*s87 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s88 :|: s85 >= 0, s85 <= x_111 + 1, s86 >= 0, s86 <= x_25 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 2007 + 2*s89 + 2*s89*s90 + 5*s90 + 2*s90^2 + 2*s91 + 2*s91*s94 + 2*s92 + 2*s92*s93 + 5*s93 + 2*s93^2 + 5*s94 + 2*s94^2 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_115 + 916*x_115^2 + 188*x_115^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 1407*x_27 + 916*x_27^2 + 188*x_27^3 }-> s95 :|: s89 >= 0, s89 <= x_111 + 1, s90 >= 0, s90 <= x_25 + 1, s91 >= 0, s91 <= s89 + s90 + 1, s92 >= 0, s92 <= x_115 + 1, s93 >= 0, s93 <= x_27 + 1, s94 >= 0, s94 <= s92 + s93 + 1, s95 >= 0, s95 <= s91 + s94 + 1, x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 1001 + 2*s130 + 2*s130*s131 + 5*s131 + 2*s131^2 + s132 + 1407*x_121 + 916*x_121^2 + 188*x_121^3 + 1407*x_210 + 916*x_210^2 + 188*x_210^3 }-> s133 :|: s130 >= 0, s130 <= x_121 + 1, s131 >= 0, s131 <= x_210 + 1, s132 >= 0, s132 <= s130 + s131 + 1, s133 >= 0, s133 <= s132 + 1, z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ -181 + s134 + s135 + 139*z + 352*z^2 + 188*z^3 }-> s136 :|: s134 >= 0, s134 <= z - 1 + 1, s135 >= 0, s135 <= s134 + 1, s136 >= 0, s136 <= s135 + 1, z - 1 >= 0
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_s}, {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^3) [498 + 1407*z + 916*z^2 + 188*z^3], size: O(n^1) [1 + z]
  encode_s: runtime: ?, size: O(n^1) [3 + z]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1013 + 2972*z + 2188*z^2 + 564*z^3

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

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

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s19 + 2*s19*s20 + 5*s20 + 2*s20^2 + 5*s21 + 2*s21^2 + 1407*x_11 + 916*x_11^2 + 188*x_11^3 + 1407*x_2'' + 916*x_2''^2 + 188*x_2''^3 }-> s22 :|: s19 >= 0, s19 <= x_11 + 1, s20 >= 0, s20 <= x_2'' + 1, s21 >= 0, s21 <= s19 + s20 + 1, s22 >= 0, s22 <= 0 + s21 + 1, z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ -151 + s23 + 5*s24 + 2*s24^2 + -1*z + -212*z^2 + 188*z^3 }-> s25 :|: s23 >= 0, s23 <= z - 2 + 1, s24 >= 0, s24 <= s23 + 1, s25 >= 0, s25 <= 0 + s24 + 1, z - 2 >= 0
   encArg(z) -{ 1006 + 2*s26 + 2*s26*s27 + 5*s27 + 2*s27^2 + 2*s28 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s29 :|: s26 >= 0, s26 <= x_1' + 1, s27 >= 0, s27 <= x_2' + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s28 + 0 + 1, x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 2007 + 2*s30 + 2*s30*s31 + 5*s31 + 2*s31^2 + 2*s32 + 2*s32*s35 + 2*s33 + 2*s33*s34 + 5*s34 + 2*s34^2 + 5*s35 + 2*s35^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_13 + 916*x_13^2 + 188*x_13^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 + 1407*x_21 + 916*x_21^2 + 188*x_21^3 }-> s36 :|: s30 >= 0, s30 <= x_1' + 1, s31 >= 0, s31 <= x_2' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= x_13 + 1, s34 >= 0, s34 <= x_21 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s32 + s35 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 1504 + 2*s37 + 2*s37*s38 + 5*s38 + 2*s38^2 + 2*s39 + 2*s39*s41 + s40 + 5*s41 + 2*s41^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_14 + 916*x_14^2 + 188*x_14^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s42 :|: s37 >= 0, s37 <= x_1' + 1, s38 >= 0, s38 <= x_2' + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= x_14 + 1, s41 >= 0, s41 <= s40 + 1, s42 >= 0, s42 <= s39 + s41 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 1006 + 2*s43 + 2*s43*s44 + 5*s44 + 2*s44^2 + 2*s45 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s46 :|: s43 >= 0, s43 <= x_1' + 1, s44 >= 0, s44 <= x_2' + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s45 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ -151 + s47 + 2*s48 + -1*z + -212*z^2 + 188*z^3 }-> s49 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= s47 + 1, s49 >= 0, s49 <= s48 + 0 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 1504 + s50 + 2*s51 + 2*s51*s54 + 2*s52 + 2*s52*s53 + 5*s53 + 2*s53^2 + 5*s54 + 2*s54^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_15 + 916*x_15^2 + 188*x_15^3 + 1407*x_22 + 916*x_22^2 + 188*x_22^3 }-> s55 :|: s50 >= 0, s50 <= x_1'' + 1, s51 >= 0, s51 <= s50 + 1, s52 >= 0, s52 <= x_15 + 1, s53 >= 0, s53 <= x_22 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s51 + s54 + 1, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1001 + s56 + 2*s57 + 2*s57*s59 + s58 + 5*s59 + 2*s59^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_16 + 916*x_16^2 + 188*x_16^3 }-> s60 :|: s56 >= 0, s56 <= x_1'' + 1, s57 >= 0, s57 <= s56 + 1, s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s57 + s59 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 503 + s61 + 2*s62 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 }-> s63 :|: s61 >= 0, s61 <= x_1'' + 1, s62 >= 0, s62 <= s61 + 1, s63 >= 0, s63 <= s62 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s64 + 2*s64*s65 + 5*s65 + 2*s65^2 + 5*s66 + 2*s66^2 + 1407*x_17 + 916*x_17^2 + 188*x_17^3 + 1407*x_23 + 916*x_23^2 + 188*x_23^3 }-> s67 :|: s64 >= 0, s64 <= x_17 + 1, s65 >= 0, s65 <= x_23 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= 0 + s66 + 1, x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 503 + s68 + 5*s69 + 2*s69^2 + 1407*x_18 + 916*x_18^2 + 188*x_18^3 }-> s70 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= s68 + 1, s70 >= 0, s70 <= 0 + s69 + 1, x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 1001 + 2*s71 + 2*s71*s72 + 5*s72 + 2*s72^2 + s73 + 1407*x_19 + 916*x_19^2 + 188*x_19^3 + 1407*x_24 + 916*x_24^2 + 188*x_24^3 }-> s74 :|: s71 >= 0, s71 <= x_19 + 1, s72 >= 0, s72 <= x_24 + 1, s73 >= 0, s73 <= s71 + s72 + 1, s74 >= 0, s74 <= s73 + 1, z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ -156 + s75 + s76 + -1*z + -212*z^2 + 188*z^3 }-> s77 :|: s75 >= 0, s75 <= z - 2 + 1, s76 >= 0, s76 <= s75 + 1, s77 >= 0, s77 <= s76 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 825 + 5*s100 + 2*s100*s98 + 2*s100^2 + 2*s96 + 2*s96*s97 + 5*s97 + 2*s97^2 + 2*s98 + s99 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s101 :|: s96 >= 0, s96 <= x_111 + 1, s97 >= 0, s97 <= x_25 + 1, s98 >= 0, s98 <= s96 + s97 + 1, s99 >= 0, s99 <= z' - 1 + 1, s100 >= 0, s100 <= s99 + 1, s101 >= 0, s101 <= s98 + s100 + 1, z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 1006 + 2*s102 + 2*s102*s103 + 5*s103 + 2*s103^2 + 2*s104 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s105 :|: s102 >= 0, s102 <= x_111 + 1, s103 >= 0, s103 <= x_25 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= s104 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ -176 + s106 + 2*s107 + 139*z + 352*z^2 + 188*z^3 }-> s108 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= s106 + 1, s108 >= 0, s108 <= s107 + 0 + 1, z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 825 + s109 + 2*s110 + 2*s110*s113 + 2*s111 + 2*s111*s112 + 5*s112 + 2*s112^2 + 5*s113 + 2*s113^2 + 1407*x_117 + 916*x_117^2 + 188*x_117^3 + 1407*x_28 + 916*x_28^2 + 188*x_28^3 + 139*z + 352*z^2 + 188*z^3 }-> s114 :|: s109 >= 0, s109 <= z - 1 + 1, s110 >= 0, s110 <= s109 + 1, s111 >= 0, s111 <= x_117 + 1, s112 >= 0, s112 <= x_28 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s110 + s113 + 1, x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ -357 + s115 + 2*s116 + 2*s116*s118 + s117 + 5*s118 + 2*s118^2 + 139*z + 352*z^2 + 188*z^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s119 :|: s115 >= 0, s115 <= z - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= z' - 1 + 1, s118 >= 0, s118 <= s117 + 1, s119 >= 0, s119 <= s116 + s118 + 1, z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ -176 + s120 + 2*s121 + 139*z + 352*z^2 + 188*z^3 }-> s122 :|: s120 >= 0, s120 <= z - 1 + 1, s121 >= 0, s121 <= s120 + 1, s122 >= 0, s122 <= s121 + 0 + 1, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 1006 + 2*s123 + 2*s123*s124 + 5*s124 + 2*s124^2 + 5*s125 + 2*s125^2 + 1407*x_119 + 916*x_119^2 + 188*x_119^3 + 1407*x_29 + 916*x_29^2 + 188*x_29^3 }-> s126 :|: s123 >= 0, s123 <= x_119 + 1, s124 >= 0, s124 <= x_29 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= 0 + s125 + 1, z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ -176 + s127 + 5*s128 + 2*s128^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s129 :|: s127 >= 0, s127 <= z' - 1 + 1, s128 >= 0, s128 <= s127 + 1, s129 >= 0, s129 <= 0 + s128 + 1, z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 1006 + 2*s78 + 2*s78*s79 + 5*s79 + 2*s79^2 + 5*s80 + 2*s80^2 + 1407*x_113 + 916*x_113^2 + 188*x_113^3 + 1407*x_26 + 916*x_26^2 + 188*x_26^3 }-> s81 :|: s78 >= 0, s78 <= x_113 + 1, s79 >= 0, s79 <= x_26 + 1, s80 >= 0, s80 <= s78 + s79 + 1, s81 >= 0, s81 <= 0 + s80 + 1, x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ -176 + s82 + 5*s83 + 2*s83^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s84 :|: s82 >= 0, s82 <= z' - 1 + 1, s83 >= 0, s83 <= s82 + 1, s84 >= 0, s84 <= 0 + s83 + 1, z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 1006 + 2*s85 + 2*s85*s86 + 5*s86 + 2*s86^2 + 2*s87 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s88 :|: s85 >= 0, s85 <= x_111 + 1, s86 >= 0, s86 <= x_25 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 2007 + 2*s89 + 2*s89*s90 + 5*s90 + 2*s90^2 + 2*s91 + 2*s91*s94 + 2*s92 + 2*s92*s93 + 5*s93 + 2*s93^2 + 5*s94 + 2*s94^2 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_115 + 916*x_115^2 + 188*x_115^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 1407*x_27 + 916*x_27^2 + 188*x_27^3 }-> s95 :|: s89 >= 0, s89 <= x_111 + 1, s90 >= 0, s90 <= x_25 + 1, s91 >= 0, s91 <= s89 + s90 + 1, s92 >= 0, s92 <= x_115 + 1, s93 >= 0, s93 <= x_27 + 1, s94 >= 0, s94 <= s92 + s93 + 1, s95 >= 0, s95 <= s91 + s94 + 1, x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 1001 + 2*s130 + 2*s130*s131 + 5*s131 + 2*s131^2 + s132 + 1407*x_121 + 916*x_121^2 + 188*x_121^3 + 1407*x_210 + 916*x_210^2 + 188*x_210^3 }-> s133 :|: s130 >= 0, s130 <= x_121 + 1, s131 >= 0, s131 <= x_210 + 1, s132 >= 0, s132 <= s130 + s131 + 1, s133 >= 0, s133 <= s132 + 1, z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ -181 + s134 + s135 + 139*z + 352*z^2 + 188*z^3 }-> s136 :|: s134 >= 0, s134 <= z - 1 + 1, s135 >= 0, s135 <= s134 + 1, s136 >= 0, s136 <= s135 + 1, z - 1 >= 0
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^3) [498 + 1407*z + 916*z^2 + 188*z^3], size: O(n^1) [1 + z]
  encode_s: runtime: O(n^3) [1013 + 2972*z + 2188*z^2 + 564*z^3], size: O(n^1) [3 + 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:

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s19 + 2*s19*s20 + 5*s20 + 2*s20^2 + 5*s21 + 2*s21^2 + 1407*x_11 + 916*x_11^2 + 188*x_11^3 + 1407*x_2'' + 916*x_2''^2 + 188*x_2''^3 }-> s22 :|: s19 >= 0, s19 <= x_11 + 1, s20 >= 0, s20 <= x_2'' + 1, s21 >= 0, s21 <= s19 + s20 + 1, s22 >= 0, s22 <= 0 + s21 + 1, z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ -151 + s23 + 5*s24 + 2*s24^2 + -1*z + -212*z^2 + 188*z^3 }-> s25 :|: s23 >= 0, s23 <= z - 2 + 1, s24 >= 0, s24 <= s23 + 1, s25 >= 0, s25 <= 0 + s24 + 1, z - 2 >= 0
   encArg(z) -{ 1006 + 2*s26 + 2*s26*s27 + 5*s27 + 2*s27^2 + 2*s28 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s29 :|: s26 >= 0, s26 <= x_1' + 1, s27 >= 0, s27 <= x_2' + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s28 + 0 + 1, x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 2007 + 2*s30 + 2*s30*s31 + 5*s31 + 2*s31^2 + 2*s32 + 2*s32*s35 + 2*s33 + 2*s33*s34 + 5*s34 + 2*s34^2 + 5*s35 + 2*s35^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_13 + 916*x_13^2 + 188*x_13^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 + 1407*x_21 + 916*x_21^2 + 188*x_21^3 }-> s36 :|: s30 >= 0, s30 <= x_1' + 1, s31 >= 0, s31 <= x_2' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= x_13 + 1, s34 >= 0, s34 <= x_21 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s32 + s35 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 1504 + 2*s37 + 2*s37*s38 + 5*s38 + 2*s38^2 + 2*s39 + 2*s39*s41 + s40 + 5*s41 + 2*s41^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_14 + 916*x_14^2 + 188*x_14^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s42 :|: s37 >= 0, s37 <= x_1' + 1, s38 >= 0, s38 <= x_2' + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= x_14 + 1, s41 >= 0, s41 <= s40 + 1, s42 >= 0, s42 <= s39 + s41 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 1006 + 2*s43 + 2*s43*s44 + 5*s44 + 2*s44^2 + 2*s45 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s46 :|: s43 >= 0, s43 <= x_1' + 1, s44 >= 0, s44 <= x_2' + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s45 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ -151 + s47 + 2*s48 + -1*z + -212*z^2 + 188*z^3 }-> s49 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= s47 + 1, s49 >= 0, s49 <= s48 + 0 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 1504 + s50 + 2*s51 + 2*s51*s54 + 2*s52 + 2*s52*s53 + 5*s53 + 2*s53^2 + 5*s54 + 2*s54^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_15 + 916*x_15^2 + 188*x_15^3 + 1407*x_22 + 916*x_22^2 + 188*x_22^3 }-> s55 :|: s50 >= 0, s50 <= x_1'' + 1, s51 >= 0, s51 <= s50 + 1, s52 >= 0, s52 <= x_15 + 1, s53 >= 0, s53 <= x_22 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s51 + s54 + 1, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1001 + s56 + 2*s57 + 2*s57*s59 + s58 + 5*s59 + 2*s59^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_16 + 916*x_16^2 + 188*x_16^3 }-> s60 :|: s56 >= 0, s56 <= x_1'' + 1, s57 >= 0, s57 <= s56 + 1, s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s57 + s59 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 503 + s61 + 2*s62 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 }-> s63 :|: s61 >= 0, s61 <= x_1'' + 1, s62 >= 0, s62 <= s61 + 1, s63 >= 0, s63 <= s62 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s64 + 2*s64*s65 + 5*s65 + 2*s65^2 + 5*s66 + 2*s66^2 + 1407*x_17 + 916*x_17^2 + 188*x_17^3 + 1407*x_23 + 916*x_23^2 + 188*x_23^3 }-> s67 :|: s64 >= 0, s64 <= x_17 + 1, s65 >= 0, s65 <= x_23 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= 0 + s66 + 1, x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 503 + s68 + 5*s69 + 2*s69^2 + 1407*x_18 + 916*x_18^2 + 188*x_18^3 }-> s70 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= s68 + 1, s70 >= 0, s70 <= 0 + s69 + 1, x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 1001 + 2*s71 + 2*s71*s72 + 5*s72 + 2*s72^2 + s73 + 1407*x_19 + 916*x_19^2 + 188*x_19^3 + 1407*x_24 + 916*x_24^2 + 188*x_24^3 }-> s74 :|: s71 >= 0, s71 <= x_19 + 1, s72 >= 0, s72 <= x_24 + 1, s73 >= 0, s73 <= s71 + s72 + 1, s74 >= 0, s74 <= s73 + 1, z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ -156 + s75 + s76 + -1*z + -212*z^2 + 188*z^3 }-> s77 :|: s75 >= 0, s75 <= z - 2 + 1, s76 >= 0, s76 <= s75 + 1, s77 >= 0, s77 <= s76 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 825 + 5*s100 + 2*s100*s98 + 2*s100^2 + 2*s96 + 2*s96*s97 + 5*s97 + 2*s97^2 + 2*s98 + s99 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s101 :|: s96 >= 0, s96 <= x_111 + 1, s97 >= 0, s97 <= x_25 + 1, s98 >= 0, s98 <= s96 + s97 + 1, s99 >= 0, s99 <= z' - 1 + 1, s100 >= 0, s100 <= s99 + 1, s101 >= 0, s101 <= s98 + s100 + 1, z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 1006 + 2*s102 + 2*s102*s103 + 5*s103 + 2*s103^2 + 2*s104 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s105 :|: s102 >= 0, s102 <= x_111 + 1, s103 >= 0, s103 <= x_25 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= s104 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ -176 + s106 + 2*s107 + 139*z + 352*z^2 + 188*z^3 }-> s108 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= s106 + 1, s108 >= 0, s108 <= s107 + 0 + 1, z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 825 + s109 + 2*s110 + 2*s110*s113 + 2*s111 + 2*s111*s112 + 5*s112 + 2*s112^2 + 5*s113 + 2*s113^2 + 1407*x_117 + 916*x_117^2 + 188*x_117^3 + 1407*x_28 + 916*x_28^2 + 188*x_28^3 + 139*z + 352*z^2 + 188*z^3 }-> s114 :|: s109 >= 0, s109 <= z - 1 + 1, s110 >= 0, s110 <= s109 + 1, s111 >= 0, s111 <= x_117 + 1, s112 >= 0, s112 <= x_28 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s110 + s113 + 1, x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ -357 + s115 + 2*s116 + 2*s116*s118 + s117 + 5*s118 + 2*s118^2 + 139*z + 352*z^2 + 188*z^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s119 :|: s115 >= 0, s115 <= z - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= z' - 1 + 1, s118 >= 0, s118 <= s117 + 1, s119 >= 0, s119 <= s116 + s118 + 1, z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ -176 + s120 + 2*s121 + 139*z + 352*z^2 + 188*z^3 }-> s122 :|: s120 >= 0, s120 <= z - 1 + 1, s121 >= 0, s121 <= s120 + 1, s122 >= 0, s122 <= s121 + 0 + 1, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 1006 + 2*s123 + 2*s123*s124 + 5*s124 + 2*s124^2 + 5*s125 + 2*s125^2 + 1407*x_119 + 916*x_119^2 + 188*x_119^3 + 1407*x_29 + 916*x_29^2 + 188*x_29^3 }-> s126 :|: s123 >= 0, s123 <= x_119 + 1, s124 >= 0, s124 <= x_29 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= 0 + s125 + 1, z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ -176 + s127 + 5*s128 + 2*s128^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s129 :|: s127 >= 0, s127 <= z' - 1 + 1, s128 >= 0, s128 <= s127 + 1, s129 >= 0, s129 <= 0 + s128 + 1, z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 1006 + 2*s78 + 2*s78*s79 + 5*s79 + 2*s79^2 + 5*s80 + 2*s80^2 + 1407*x_113 + 916*x_113^2 + 188*x_113^3 + 1407*x_26 + 916*x_26^2 + 188*x_26^3 }-> s81 :|: s78 >= 0, s78 <= x_113 + 1, s79 >= 0, s79 <= x_26 + 1, s80 >= 0, s80 <= s78 + s79 + 1, s81 >= 0, s81 <= 0 + s80 + 1, x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ -176 + s82 + 5*s83 + 2*s83^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s84 :|: s82 >= 0, s82 <= z' - 1 + 1, s83 >= 0, s83 <= s82 + 1, s84 >= 0, s84 <= 0 + s83 + 1, z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 1006 + 2*s85 + 2*s85*s86 + 5*s86 + 2*s86^2 + 2*s87 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s88 :|: s85 >= 0, s85 <= x_111 + 1, s86 >= 0, s86 <= x_25 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 2007 + 2*s89 + 2*s89*s90 + 5*s90 + 2*s90^2 + 2*s91 + 2*s91*s94 + 2*s92 + 2*s92*s93 + 5*s93 + 2*s93^2 + 5*s94 + 2*s94^2 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_115 + 916*x_115^2 + 188*x_115^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 1407*x_27 + 916*x_27^2 + 188*x_27^3 }-> s95 :|: s89 >= 0, s89 <= x_111 + 1, s90 >= 0, s90 <= x_25 + 1, s91 >= 0, s91 <= s89 + s90 + 1, s92 >= 0, s92 <= x_115 + 1, s93 >= 0, s93 <= x_27 + 1, s94 >= 0, s94 <= s92 + s93 + 1, s95 >= 0, s95 <= s91 + s94 + 1, x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 1001 + 2*s130 + 2*s130*s131 + 5*s131 + 2*s131^2 + s132 + 1407*x_121 + 916*x_121^2 + 188*x_121^3 + 1407*x_210 + 916*x_210^2 + 188*x_210^3 }-> s133 :|: s130 >= 0, s130 <= x_121 + 1, s131 >= 0, s131 <= x_210 + 1, s132 >= 0, s132 <= s130 + s131 + 1, s133 >= 0, s133 <= s132 + 1, z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ -181 + s134 + s135 + 139*z + 352*z^2 + 188*z^3 }-> s136 :|: s134 >= 0, s134 <= z - 1 + 1, s135 >= 0, s135 <= s134 + 1, s136 >= 0, s136 <= s135 + 1, z - 1 >= 0
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^3) [498 + 1407*z + 916*z^2 + 188*z^3], size: O(n^1) [1 + z]
  encode_s: runtime: O(n^3) [1013 + 2972*z + 2188*z^2 + 564*z^3], size: O(n^1) [3 + z]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s19 + 2*s19*s20 + 5*s20 + 2*s20^2 + 5*s21 + 2*s21^2 + 1407*x_11 + 916*x_11^2 + 188*x_11^3 + 1407*x_2'' + 916*x_2''^2 + 188*x_2''^3 }-> s22 :|: s19 >= 0, s19 <= x_11 + 1, s20 >= 0, s20 <= x_2'' + 1, s21 >= 0, s21 <= s19 + s20 + 1, s22 >= 0, s22 <= 0 + s21 + 1, z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ -151 + s23 + 5*s24 + 2*s24^2 + -1*z + -212*z^2 + 188*z^3 }-> s25 :|: s23 >= 0, s23 <= z - 2 + 1, s24 >= 0, s24 <= s23 + 1, s25 >= 0, s25 <= 0 + s24 + 1, z - 2 >= 0
   encArg(z) -{ 1006 + 2*s26 + 2*s26*s27 + 5*s27 + 2*s27^2 + 2*s28 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s29 :|: s26 >= 0, s26 <= x_1' + 1, s27 >= 0, s27 <= x_2' + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s28 + 0 + 1, x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 2007 + 2*s30 + 2*s30*s31 + 5*s31 + 2*s31^2 + 2*s32 + 2*s32*s35 + 2*s33 + 2*s33*s34 + 5*s34 + 2*s34^2 + 5*s35 + 2*s35^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_13 + 916*x_13^2 + 188*x_13^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 + 1407*x_21 + 916*x_21^2 + 188*x_21^3 }-> s36 :|: s30 >= 0, s30 <= x_1' + 1, s31 >= 0, s31 <= x_2' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= x_13 + 1, s34 >= 0, s34 <= x_21 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s32 + s35 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 1504 + 2*s37 + 2*s37*s38 + 5*s38 + 2*s38^2 + 2*s39 + 2*s39*s41 + s40 + 5*s41 + 2*s41^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_14 + 916*x_14^2 + 188*x_14^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s42 :|: s37 >= 0, s37 <= x_1' + 1, s38 >= 0, s38 <= x_2' + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= x_14 + 1, s41 >= 0, s41 <= s40 + 1, s42 >= 0, s42 <= s39 + s41 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 1006 + 2*s43 + 2*s43*s44 + 5*s44 + 2*s44^2 + 2*s45 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s46 :|: s43 >= 0, s43 <= x_1' + 1, s44 >= 0, s44 <= x_2' + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s45 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ -151 + s47 + 2*s48 + -1*z + -212*z^2 + 188*z^3 }-> s49 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= s47 + 1, s49 >= 0, s49 <= s48 + 0 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 1504 + s50 + 2*s51 + 2*s51*s54 + 2*s52 + 2*s52*s53 + 5*s53 + 2*s53^2 + 5*s54 + 2*s54^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_15 + 916*x_15^2 + 188*x_15^3 + 1407*x_22 + 916*x_22^2 + 188*x_22^3 }-> s55 :|: s50 >= 0, s50 <= x_1'' + 1, s51 >= 0, s51 <= s50 + 1, s52 >= 0, s52 <= x_15 + 1, s53 >= 0, s53 <= x_22 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s51 + s54 + 1, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1001 + s56 + 2*s57 + 2*s57*s59 + s58 + 5*s59 + 2*s59^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_16 + 916*x_16^2 + 188*x_16^3 }-> s60 :|: s56 >= 0, s56 <= x_1'' + 1, s57 >= 0, s57 <= s56 + 1, s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s57 + s59 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 503 + s61 + 2*s62 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 }-> s63 :|: s61 >= 0, s61 <= x_1'' + 1, s62 >= 0, s62 <= s61 + 1, s63 >= 0, s63 <= s62 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s64 + 2*s64*s65 + 5*s65 + 2*s65^2 + 5*s66 + 2*s66^2 + 1407*x_17 + 916*x_17^2 + 188*x_17^3 + 1407*x_23 + 916*x_23^2 + 188*x_23^3 }-> s67 :|: s64 >= 0, s64 <= x_17 + 1, s65 >= 0, s65 <= x_23 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= 0 + s66 + 1, x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 503 + s68 + 5*s69 + 2*s69^2 + 1407*x_18 + 916*x_18^2 + 188*x_18^3 }-> s70 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= s68 + 1, s70 >= 0, s70 <= 0 + s69 + 1, x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 1001 + 2*s71 + 2*s71*s72 + 5*s72 + 2*s72^2 + s73 + 1407*x_19 + 916*x_19^2 + 188*x_19^3 + 1407*x_24 + 916*x_24^2 + 188*x_24^3 }-> s74 :|: s71 >= 0, s71 <= x_19 + 1, s72 >= 0, s72 <= x_24 + 1, s73 >= 0, s73 <= s71 + s72 + 1, s74 >= 0, s74 <= s73 + 1, z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ -156 + s75 + s76 + -1*z + -212*z^2 + 188*z^3 }-> s77 :|: s75 >= 0, s75 <= z - 2 + 1, s76 >= 0, s76 <= s75 + 1, s77 >= 0, s77 <= s76 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 825 + 5*s100 + 2*s100*s98 + 2*s100^2 + 2*s96 + 2*s96*s97 + 5*s97 + 2*s97^2 + 2*s98 + s99 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s101 :|: s96 >= 0, s96 <= x_111 + 1, s97 >= 0, s97 <= x_25 + 1, s98 >= 0, s98 <= s96 + s97 + 1, s99 >= 0, s99 <= z' - 1 + 1, s100 >= 0, s100 <= s99 + 1, s101 >= 0, s101 <= s98 + s100 + 1, z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 1006 + 2*s102 + 2*s102*s103 + 5*s103 + 2*s103^2 + 2*s104 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s105 :|: s102 >= 0, s102 <= x_111 + 1, s103 >= 0, s103 <= x_25 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= s104 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ -176 + s106 + 2*s107 + 139*z + 352*z^2 + 188*z^3 }-> s108 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= s106 + 1, s108 >= 0, s108 <= s107 + 0 + 1, z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 825 + s109 + 2*s110 + 2*s110*s113 + 2*s111 + 2*s111*s112 + 5*s112 + 2*s112^2 + 5*s113 + 2*s113^2 + 1407*x_117 + 916*x_117^2 + 188*x_117^3 + 1407*x_28 + 916*x_28^2 + 188*x_28^3 + 139*z + 352*z^2 + 188*z^3 }-> s114 :|: s109 >= 0, s109 <= z - 1 + 1, s110 >= 0, s110 <= s109 + 1, s111 >= 0, s111 <= x_117 + 1, s112 >= 0, s112 <= x_28 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s110 + s113 + 1, x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ -357 + s115 + 2*s116 + 2*s116*s118 + s117 + 5*s118 + 2*s118^2 + 139*z + 352*z^2 + 188*z^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s119 :|: s115 >= 0, s115 <= z - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= z' - 1 + 1, s118 >= 0, s118 <= s117 + 1, s119 >= 0, s119 <= s116 + s118 + 1, z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ -176 + s120 + 2*s121 + 139*z + 352*z^2 + 188*z^3 }-> s122 :|: s120 >= 0, s120 <= z - 1 + 1, s121 >= 0, s121 <= s120 + 1, s122 >= 0, s122 <= s121 + 0 + 1, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 1006 + 2*s123 + 2*s123*s124 + 5*s124 + 2*s124^2 + 5*s125 + 2*s125^2 + 1407*x_119 + 916*x_119^2 + 188*x_119^3 + 1407*x_29 + 916*x_29^2 + 188*x_29^3 }-> s126 :|: s123 >= 0, s123 <= x_119 + 1, s124 >= 0, s124 <= x_29 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= 0 + s125 + 1, z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ -176 + s127 + 5*s128 + 2*s128^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s129 :|: s127 >= 0, s127 <= z' - 1 + 1, s128 >= 0, s128 <= s127 + 1, s129 >= 0, s129 <= 0 + s128 + 1, z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 1006 + 2*s78 + 2*s78*s79 + 5*s79 + 2*s79^2 + 5*s80 + 2*s80^2 + 1407*x_113 + 916*x_113^2 + 188*x_113^3 + 1407*x_26 + 916*x_26^2 + 188*x_26^3 }-> s81 :|: s78 >= 0, s78 <= x_113 + 1, s79 >= 0, s79 <= x_26 + 1, s80 >= 0, s80 <= s78 + s79 + 1, s81 >= 0, s81 <= 0 + s80 + 1, x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ -176 + s82 + 5*s83 + 2*s83^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s84 :|: s82 >= 0, s82 <= z' - 1 + 1, s83 >= 0, s83 <= s82 + 1, s84 >= 0, s84 <= 0 + s83 + 1, z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 1006 + 2*s85 + 2*s85*s86 + 5*s86 + 2*s86^2 + 2*s87 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s88 :|: s85 >= 0, s85 <= x_111 + 1, s86 >= 0, s86 <= x_25 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 2007 + 2*s89 + 2*s89*s90 + 5*s90 + 2*s90^2 + 2*s91 + 2*s91*s94 + 2*s92 + 2*s92*s93 + 5*s93 + 2*s93^2 + 5*s94 + 2*s94^2 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_115 + 916*x_115^2 + 188*x_115^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 1407*x_27 + 916*x_27^2 + 188*x_27^3 }-> s95 :|: s89 >= 0, s89 <= x_111 + 1, s90 >= 0, s90 <= x_25 + 1, s91 >= 0, s91 <= s89 + s90 + 1, s92 >= 0, s92 <= x_115 + 1, s93 >= 0, s93 <= x_27 + 1, s94 >= 0, s94 <= s92 + s93 + 1, s95 >= 0, s95 <= s91 + s94 + 1, x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 1001 + 2*s130 + 2*s130*s131 + 5*s131 + 2*s131^2 + s132 + 1407*x_121 + 916*x_121^2 + 188*x_121^3 + 1407*x_210 + 916*x_210^2 + 188*x_210^3 }-> s133 :|: s130 >= 0, s130 <= x_121 + 1, s131 >= 0, s131 <= x_210 + 1, s132 >= 0, s132 <= s130 + s131 + 1, s133 >= 0, s133 <= s132 + 1, z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ -181 + s134 + s135 + 139*z + 352*z^2 + 188*z^3 }-> s136 :|: s134 >= 0, s134 <= z - 1 + 1, s135 >= 0, s135 <= s134 + 1, s136 >= 0, s136 <= s135 + 1, z - 1 >= 0
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {encode_+}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^3) [498 + 1407*z + 916*z^2 + 188*z^3], size: O(n^1) [1 + z]
  encode_s: runtime: O(n^3) [1013 + 2972*z + 2188*z^2 + 564*z^3], size: O(n^1) [3 + z]
  encode_+: runtime: ?, size: O(n^1) [5 + z + z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_+
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 7840 + 11912*z + 18*z*z' + 8752*z^2 + 2256*z^3 + 11996*z' + 8792*z'^2 + 2256*z'^3

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

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

   encArg(z) -{ 5 }-> s12 :|: s12 >= 0, s12 <= 0 + 0 + 1, z = 1 + 0 + 0
   encArg(z) -{ 5 }-> s13 :|: s13 >= 0, s13 <= 0 + 0 + 1, z - 1 >= 0
   encArg(z) -{ 5 }-> s14 :|: s14 >= 0, s14 <= 0 + 0 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s19 + 2*s19*s20 + 5*s20 + 2*s20^2 + 5*s21 + 2*s21^2 + 1407*x_11 + 916*x_11^2 + 188*x_11^3 + 1407*x_2'' + 916*x_2''^2 + 188*x_2''^3 }-> s22 :|: s19 >= 0, s19 <= x_11 + 1, s20 >= 0, s20 <= x_2'' + 1, s21 >= 0, s21 <= s19 + s20 + 1, s22 >= 0, s22 <= 0 + s21 + 1, z = 1 + 0 + (1 + x_11 + x_2''), x_11 >= 0, x_2'' >= 0
   encArg(z) -{ -151 + s23 + 5*s24 + 2*s24^2 + -1*z + -212*z^2 + 188*z^3 }-> s25 :|: s23 >= 0, s23 <= z - 2 + 1, s24 >= 0, s24 <= s23 + 1, s25 >= 0, s25 <= 0 + s24 + 1, z - 2 >= 0
   encArg(z) -{ 1006 + 2*s26 + 2*s26*s27 + 5*s27 + 2*s27^2 + 2*s28 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s29 :|: s26 >= 0, s26 <= x_1' + 1, s27 >= 0, s27 <= x_2' + 1, s28 >= 0, s28 <= s26 + s27 + 1, s29 >= 0, s29 <= s28 + 0 + 1, x_2' >= 0, x_1' >= 0, z = 1 + (1 + x_1' + x_2') + 0
   encArg(z) -{ 2007 + 2*s30 + 2*s30*s31 + 5*s31 + 2*s31^2 + 2*s32 + 2*s32*s35 + 2*s33 + 2*s33*s34 + 5*s34 + 2*s34^2 + 5*s35 + 2*s35^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_13 + 916*x_13^2 + 188*x_13^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 + 1407*x_21 + 916*x_21^2 + 188*x_21^3 }-> s36 :|: s30 >= 0, s30 <= x_1' + 1, s31 >= 0, s31 <= x_2' + 1, s32 >= 0, s32 <= s30 + s31 + 1, s33 >= 0, s33 <= x_13 + 1, s34 >= 0, s34 <= x_21 + 1, s35 >= 0, s35 <= s33 + s34 + 1, s36 >= 0, s36 <= s32 + s35 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_13 + x_21), x_2' >= 0, x_13 >= 0, x_1' >= 0, x_21 >= 0
   encArg(z) -{ 1504 + 2*s37 + 2*s37*s38 + 5*s38 + 2*s38^2 + 2*s39 + 2*s39*s41 + s40 + 5*s41 + 2*s41^2 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_14 + 916*x_14^2 + 188*x_14^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s42 :|: s37 >= 0, s37 <= x_1' + 1, s38 >= 0, s38 <= x_2' + 1, s39 >= 0, s39 <= s37 + s38 + 1, s40 >= 0, s40 <= x_14 + 1, s41 >= 0, s41 <= s40 + 1, s42 >= 0, s42 <= s39 + s41 + 1, z = 1 + (1 + x_1' + x_2') + (1 + x_14), x_14 >= 0, x_2' >= 0, x_1' >= 0
   encArg(z) -{ 1006 + 2*s43 + 2*s43*s44 + 5*s44 + 2*s44^2 + 2*s45 + 1407*x_1' + 916*x_1'^2 + 188*x_1'^3 + 1407*x_2' + 916*x_2'^2 + 188*x_2'^3 }-> s46 :|: s43 >= 0, s43 <= x_1' + 1, s44 >= 0, s44 <= x_2' + 1, s45 >= 0, s45 <= s43 + s44 + 1, s46 >= 0, s46 <= s45 + 0 + 1, z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ -151 + s47 + 2*s48 + -1*z + -212*z^2 + 188*z^3 }-> s49 :|: s47 >= 0, s47 <= z - 2 + 1, s48 >= 0, s48 <= s47 + 1, s49 >= 0, s49 <= s48 + 0 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0 + 1, z = 1 + 0
   encArg(z) -{ 1504 + s50 + 2*s51 + 2*s51*s54 + 2*s52 + 2*s52*s53 + 5*s53 + 2*s53^2 + 5*s54 + 2*s54^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_15 + 916*x_15^2 + 188*x_15^3 + 1407*x_22 + 916*x_22^2 + 188*x_22^3 }-> s55 :|: s50 >= 0, s50 <= x_1'' + 1, s51 >= 0, s51 <= s50 + 1, s52 >= 0, s52 <= x_15 + 1, s53 >= 0, s53 <= x_22 + 1, s54 >= 0, s54 <= s52 + s53 + 1, s55 >= 0, s55 <= s51 + s54 + 1, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> s6 :|: s6 >= 0, s6 <= 0 + 1, z - 1 >= 0
   encArg(z) -{ 1001 + s56 + 2*s57 + 2*s57*s59 + s58 + 5*s59 + 2*s59^2 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 + 1407*x_16 + 916*x_16^2 + 188*x_16^3 }-> s60 :|: s56 >= 0, s56 <= x_1'' + 1, s57 >= 0, s57 <= s56 + 1, s58 >= 0, s58 <= x_16 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s57 + s59 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 503 + s61 + 2*s62 + 1407*x_1'' + 916*x_1''^2 + 188*x_1''^3 }-> s63 :|: s61 >= 0, s61 <= x_1'' + 1, s62 >= 0, s62 <= s61 + 1, s63 >= 0, s63 <= s62 + 0 + 1, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 1006 + 2*s64 + 2*s64*s65 + 5*s65 + 2*s65^2 + 5*s66 + 2*s66^2 + 1407*x_17 + 916*x_17^2 + 188*x_17^3 + 1407*x_23 + 916*x_23^2 + 188*x_23^3 }-> s67 :|: s64 >= 0, s64 <= x_17 + 1, s65 >= 0, s65 <= x_23 + 1, s66 >= 0, s66 <= s64 + s65 + 1, s67 >= 0, s67 <= 0 + s66 + 1, x_1 >= 0, x_17 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_17 + x_23)
   encArg(z) -{ 503 + s68 + 5*s69 + 2*s69^2 + 1407*x_18 + 916*x_18^2 + 188*x_18^3 }-> s70 :|: s68 >= 0, s68 <= x_18 + 1, s69 >= 0, s69 <= s68 + 1, s70 >= 0, s70 <= 0 + s69 + 1, x_1 >= 0, z = 1 + x_1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 1001 + 2*s71 + 2*s71*s72 + 5*s72 + 2*s72^2 + s73 + 1407*x_19 + 916*x_19^2 + 188*x_19^3 + 1407*x_24 + 916*x_24^2 + 188*x_24^3 }-> s74 :|: s71 >= 0, s71 <= x_19 + 1, s72 >= 0, s72 <= x_24 + 1, s73 >= 0, s73 <= s71 + s72 + 1, s74 >= 0, s74 <= s73 + 1, z = 1 + (1 + x_19 + x_24), x_24 >= 0, x_19 >= 0
   encArg(z) -{ -156 + s75 + s76 + -1*z + -212*z^2 + 188*z^3 }-> s77 :|: s75 >= 0, s75 <= z - 2 + 1, s76 >= 0, s76 <= s75 + 1, s77 >= 0, s77 <= s76 + 1, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_+(z, z') -{ 825 + 5*s100 + 2*s100*s98 + 2*s100^2 + 2*s96 + 2*s96*s97 + 5*s97 + 2*s97^2 + 2*s98 + s99 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s101 :|: s96 >= 0, s96 <= x_111 + 1, s97 >= 0, s97 <= x_25 + 1, s98 >= 0, s98 <= s96 + s97 + 1, s99 >= 0, s99 <= z' - 1 + 1, s100 >= 0, s100 <= s99 + 1, s101 >= 0, s101 <= s98 + s100 + 1, z' - 1 >= 0, x_25 >= 0, z = 1 + x_111 + x_25, x_111 >= 0
   encode_+(z, z') -{ 1006 + 2*s102 + 2*s102*s103 + 5*s103 + 2*s103^2 + 2*s104 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s105 :|: s102 >= 0, s102 <= x_111 + 1, s103 >= 0, s103 <= x_25 + 1, s104 >= 0, s104 <= s102 + s103 + 1, s105 >= 0, s105 <= s104 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' >= 0, x_111 >= 0
   encode_+(z, z') -{ -176 + s106 + 2*s107 + 139*z + 352*z^2 + 188*z^3 }-> s108 :|: s106 >= 0, s106 <= z - 1 + 1, s107 >= 0, s107 <= s106 + 1, s108 >= 0, s108 <= s107 + 0 + 1, z - 1 >= 0, z' = 0
   encode_+(z, z') -{ 825 + s109 + 2*s110 + 2*s110*s113 + 2*s111 + 2*s111*s112 + 5*s112 + 2*s112^2 + 5*s113 + 2*s113^2 + 1407*x_117 + 916*x_117^2 + 188*x_117^3 + 1407*x_28 + 916*x_28^2 + 188*x_28^3 + 139*z + 352*z^2 + 188*z^3 }-> s114 :|: s109 >= 0, s109 <= z - 1 + 1, s110 >= 0, s110 <= s109 + 1, s111 >= 0, s111 <= x_117 + 1, s112 >= 0, s112 <= x_28 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s110 + s113 + 1, x_117 >= 0, z' = 1 + x_117 + x_28, z - 1 >= 0, x_28 >= 0
   encode_+(z, z') -{ -357 + s115 + 2*s116 + 2*s116*s118 + s117 + 5*s118 + 2*s118^2 + 139*z + 352*z^2 + 188*z^3 + 139*z' + 352*z'^2 + 188*z'^3 }-> s119 :|: s115 >= 0, s115 <= z - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= z' - 1 + 1, s118 >= 0, s118 <= s117 + 1, s119 >= 0, s119 <= s116 + s118 + 1, z - 1 >= 0, z' - 1 >= 0
   encode_+(z, z') -{ -176 + s120 + 2*s121 + 139*z + 352*z^2 + 188*z^3 }-> s122 :|: s120 >= 0, s120 <= z - 1 + 1, s121 >= 0, s121 <= s120 + 1, s122 >= 0, s122 <= s121 + 0 + 1, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 1006 + 2*s123 + 2*s123*s124 + 5*s124 + 2*s124^2 + 5*s125 + 2*s125^2 + 1407*x_119 + 916*x_119^2 + 188*x_119^3 + 1407*x_29 + 916*x_29^2 + 188*x_29^3 }-> s126 :|: s123 >= 0, s123 <= x_119 + 1, s124 >= 0, s124 <= x_29 + 1, s125 >= 0, s125 <= s123 + s124 + 1, s126 >= 0, s126 <= 0 + s125 + 1, z >= 0, z' = 1 + x_119 + x_29, x_119 >= 0, x_29 >= 0
   encode_+(z, z') -{ -176 + s127 + 5*s128 + 2*s128^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s129 :|: s127 >= 0, s127 <= z' - 1 + 1, s128 >= 0, s128 <= s127 + 1, s129 >= 0, s129 <= 0 + s128 + 1, z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 5 }-> s15 :|: s15 >= 0, s15 <= 0 + 0 + 1, z = 0, z' = 0
   encode_+(z, z') -{ 5 }-> s16 :|: s16 >= 0, s16 <= 0 + 0 + 1, z' >= 0, z = 0
   encode_+(z, z') -{ 5 }-> s17 :|: s17 >= 0, s17 <= 0 + 0 + 1, z >= 0, z' = 0
   encode_+(z, z') -{ 5 }-> s18 :|: s18 >= 0, s18 <= 0 + 0 + 1, z >= 0, z' >= 0
   encode_+(z, z') -{ 1006 + 2*s78 + 2*s78*s79 + 5*s79 + 2*s79^2 + 5*s80 + 2*s80^2 + 1407*x_113 + 916*x_113^2 + 188*x_113^3 + 1407*x_26 + 916*x_26^2 + 188*x_26^3 }-> s81 :|: s78 >= 0, s78 <= x_113 + 1, s79 >= 0, s79 <= x_26 + 1, s80 >= 0, s80 <= s78 + s79 + 1, s81 >= 0, s81 <= 0 + s80 + 1, x_113 >= 0, x_26 >= 0, z' = 1 + x_113 + x_26, z = 0
   encode_+(z, z') -{ -176 + s82 + 5*s83 + 2*s83^2 + 139*z' + 352*z'^2 + 188*z'^3 }-> s84 :|: s82 >= 0, s82 <= z' - 1 + 1, s83 >= 0, s83 <= s82 + 1, s84 >= 0, s84 <= 0 + s83 + 1, z' - 1 >= 0, z = 0
   encode_+(z, z') -{ 1006 + 2*s85 + 2*s85*s86 + 5*s86 + 2*s86^2 + 2*s87 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 }-> s88 :|: s85 >= 0, s85 <= x_111 + 1, s86 >= 0, s86 <= x_25 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 0 + 1, x_25 >= 0, z = 1 + x_111 + x_25, z' = 0, x_111 >= 0
   encode_+(z, z') -{ 2007 + 2*s89 + 2*s89*s90 + 5*s90 + 2*s90^2 + 2*s91 + 2*s91*s94 + 2*s92 + 2*s92*s93 + 5*s93 + 2*s93^2 + 5*s94 + 2*s94^2 + 1407*x_111 + 916*x_111^2 + 188*x_111^3 + 1407*x_115 + 916*x_115^2 + 188*x_115^3 + 1407*x_25 + 916*x_25^2 + 188*x_25^3 + 1407*x_27 + 916*x_27^2 + 188*x_27^3 }-> s95 :|: s89 >= 0, s89 <= x_111 + 1, s90 >= 0, s90 <= x_25 + 1, s91 >= 0, s91 <= s89 + s90 + 1, s92 >= 0, s92 <= x_115 + 1, s93 >= 0, s93 <= x_27 + 1, s94 >= 0, s94 <= s92 + s93 + 1, s95 >= 0, s95 <= s91 + s94 + 1, x_25 >= 0, x_115 >= 0, z = 1 + x_111 + x_25, z' = 1 + x_115 + x_27, x_27 >= 0, x_111 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 1001 + 2*s130 + 2*s130*s131 + 5*s131 + 2*s131^2 + s132 + 1407*x_121 + 916*x_121^2 + 188*x_121^3 + 1407*x_210 + 916*x_210^2 + 188*x_210^3 }-> s133 :|: s130 >= 0, s130 <= x_121 + 1, s131 >= 0, s131 <= x_210 + 1, s132 >= 0, s132 <= s130 + s131 + 1, s133 >= 0, s133 <= s132 + 1, z = 1 + x_121 + x_210, x_121 >= 0, x_210 >= 0
   encode_s(z) -{ -181 + s134 + s135 + 139*z + 352*z^2 + 188*z^3 }-> s136 :|: s134 >= 0, s134 <= z - 1 + 1, s135 >= 0, s135 <= s134 + 1, s136 >= 0, s136 <= s135 + 1, z - 1 >= 0
   encode_s(z) -{ 0 }-> s7 :|: s7 >= 0, s7 <= 0 + 1, z = 0
   encode_s(z) -{ 0 }-> s8 :|: s8 >= 0, s8 <= 0 + 1, z >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ z' }-> s :|: s >= 0, s <= z' - 1 + 1, z' - 1 >= 0, z = 0
   plus(z, z') -{ 2 + z }-> s'' :|: s'' >= 0, s'' <= z + 1, z >= 0, z' = 1 + 0
   plus(z, z') -{ 5 + s10 + s9 + -2*z + 2*z*z' + -3*z' + 2*z'^2 }-> s11 :|: s9 >= 0, s9 <= z + (z' - 2) + 1, s10 >= 0, s10 <= s9 + 1, s11 >= 0, s11 <= s10 + 1, z >= 0, z' - 2 >= 0
   plus(z, z') -{ s1 + z' }-> s2 :|: s1 >= 0, s1 <= z' - 2 + 1, s2 >= 0, s2 <= s1 + 1, z' - 2 >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> s3 :|: s3 >= 0, s3 <= 1 + z + (z' - 1) + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0
   s(z) -{ z }-> s' :|: s' >= 0, s' <= z - 1 + 1, z - 1 >= 0
   s(z) -{ 0 }-> 0 :|: z >= 0
   s(z) -{ 0 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  s: runtime: O(n^1) [z], size: O(n^1) [1 + z]
  plus: runtime: O(n^2) [5 + 2*z + 2*z*z' + 5*z' + 2*z'^2], size: O(n^1) [1 + z + z']
  encArg: runtime: O(n^3) [498 + 1407*z + 916*z^2 + 188*z^3], size: O(n^1) [1 + z]
  encode_s: runtime: O(n^3) [1013 + 2972*z + 2188*z^2 + 564*z^3], size: O(n^1) [3 + z]
  encode_+: runtime: O(n^3) [7840 + 11912*z + 18*z*z' + 8752*z^2 + 2256*z^3 + 11996*z' + 8792*z'^2 + 2256*z'^3], size: O(n^1) [5 + z + z']

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

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

(60)
BOUNDS(1, n^3)

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

(61) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(62)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   +'(0', s(y)) -> s(y)
   s(+'(0', y)) -> s(y)

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

   encArg(0') -> 0'
   encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))

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

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

(64)
Obligation:
Innermost TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(0', s(y)) -> s(y)
s(+'(0', y)) -> s(y)
encArg(0') -> 0'
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
+' :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
0' :: 0':cons_+:cons_s
s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
encArg :: 0':cons_+:cons_s -> 0':cons_+:cons_s
cons_+ :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
cons_s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
encode_+ :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
encode_0 :: 0':cons_+:cons_s
encode_s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
hole_0':cons_+:cons_s1_3 :: 0':cons_+:cons_s
gen_0':cons_+:cons_s2_3 :: Nat -> 0':cons_+:cons_s

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

(65) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', s, encArg

They will be analysed ascendingly in the following order:
s < +'
+' < encArg
s < encArg

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

(66)
Obligation:
Innermost TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(0', s(y)) -> s(y)
s(+'(0', y)) -> s(y)
encArg(0') -> 0'
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
+' :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
0' :: 0':cons_+:cons_s
s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
encArg :: 0':cons_+:cons_s -> 0':cons_+:cons_s
cons_+ :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
cons_s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
encode_+ :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
encode_0 :: 0':cons_+:cons_s
encode_s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
hole_0':cons_+:cons_s1_3 :: 0':cons_+:cons_s
gen_0':cons_+:cons_s2_3 :: Nat -> 0':cons_+:cons_s


Generator Equations:
gen_0':cons_+:cons_s2_3(0) <=> 0'
gen_0':cons_+:cons_s2_3(+(x, 1)) <=> cons_+(0', gen_0':cons_+:cons_s2_3(x))


The following defined symbols remain to be analysed:
s, +', encArg

They will be analysed ascendingly in the following order:
s < +'
+' < encArg
s < encArg

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':cons_+:cons_s2_3(n21_3)) -> gen_0':cons_+:cons_s2_3(0), rt in Omega(n21_3)

Induction Base:
encArg(gen_0':cons_+:cons_s2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':cons_+:cons_s2_3(+(n21_3, 1))) ->_R^Omega(0)
+'(encArg(0'), encArg(gen_0':cons_+:cons_s2_3(n21_3))) ->_R^Omega(0)
+'(0', encArg(gen_0':cons_+:cons_s2_3(n21_3))) ->_IH
+'(0', gen_0':cons_+:cons_s2_3(0)) ->_R^Omega(1)
0'

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(68)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(0', s(y)) -> s(y)
s(+'(0', y)) -> s(y)
encArg(0') -> 0'
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_s(x_1)) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
+' :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
0' :: 0':cons_+:cons_s
s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
encArg :: 0':cons_+:cons_s -> 0':cons_+:cons_s
cons_+ :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
cons_s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
encode_+ :: 0':cons_+:cons_s -> 0':cons_+:cons_s -> 0':cons_+:cons_s
encode_0 :: 0':cons_+:cons_s
encode_s :: 0':cons_+:cons_s -> 0':cons_+:cons_s
hole_0':cons_+:cons_s1_3 :: 0':cons_+:cons_s
gen_0':cons_+:cons_s2_3 :: Nat -> 0':cons_+:cons_s


Generator Equations:
gen_0':cons_+:cons_s2_3(0) <=> 0'
gen_0':cons_+:cons_s2_3(+(x, 1)) <=> cons_+(0', gen_0':cons_+:cons_s2_3(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

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

(70)
BOUNDS(n^1, INF)