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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 156 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, 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), 8 ms]
    (18) CpxRNTS
    (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 386 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 146 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 1895 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 1371 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1619 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 729 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 220 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) FinalProof [FINISHED, 0 ms]
    (54) BOUNDS(1, n^2)
(55) RenamingProof [BOTH BOUNDS(ID, ID), 2 ms]
(56) CpxRelTRS
    (57) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) typed CpxTrs
    (59) OrderProof [LOWER BOUND(ID), 0 ms]
    (60) typed CpxTrs
    (61) RewriteLemmaProof [LOWER BOUND(ID), 323 ms]
    (62) BEST
        (63) proven lower bound
            (64) LowerBoundPropagationProof [FINISHED, 0 ms]
            (65) BOUNDS(n^1, INF)
        (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
            (68) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

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

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

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

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

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

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

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

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

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

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

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

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

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   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
----------------------------------------

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   +(0, y) -> y [1]
   +(s(x), y) -> s(+(x, y)) [1]
   +(s(x), y) -> +(x, s(y)) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [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^2).


The TRS R consists of the following rules:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [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]

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(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [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]

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

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:

   plus_2
   encArg_1
   encode_+_2
   encode_0
   encode_s_1

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

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

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

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

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

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

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

   0 => 0

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

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

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


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

(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 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0


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

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

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

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

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

   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

(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 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

(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 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {plus}, {encArg}, {encode_+}, {encode_s}
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 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus}, {encArg}, {encode_+}, {encode_s}
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 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus}, {encArg}, {encode_+}, {encode_s}
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: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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

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

   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   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 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 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 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> plus(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_+}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + 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) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

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

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

(37) 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: 6*z

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

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

   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_+}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: ?, size: O(n^1) [6*z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 34*z + 12*z^2

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

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

   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), plus(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> plus(plus(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> plus(0, plus(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> plus(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), plus(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> plus(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> plus(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), plus(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(plus(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, plus(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), plus(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> plus(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_+}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [6*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) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 4 + s10 + 34*x_12 + 12*x_12^2 + 34*x_2'' + 12*x_2''^2 }-> s13 :|: s10 >= 0, s10 <= 6 * x_12, s11 >= 0, s11 <= 6 * x_2'', s12 >= 0, s12 <= s10 + s11, s13 >= 0, s13 <= 0 + s12, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ -17 + s14 + -14*z + 12*z^2 }-> s15 :|: s14 >= 0, s14 <= 6 * (z - 2), s15 >= 0, s15 <= 1 + s14 + 0, z - 2 >= 0
   encArg(z) -{ 4 + s16 + 34*x_1' + 12*x_1'^2 + 34*x_13 + 12*x_13^2 }-> s18 :|: s16 >= 0, s16 <= 6 * x_1', s17 >= 0, s17 <= 6 * x_13, s18 >= 0, s18 <= 1 + s16 + (1 + s17), z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 6 + s19 + s20 + 34*x_1' + 12*x_1'^2 + 34*x_14 + 12*x_14^2 + 34*x_21 + 12*x_21^2 }-> s23 :|: s19 >= 0, s19 <= 6 * x_1', s20 >= 0, s20 <= 6 * x_14, s21 >= 0, s21 <= 6 * x_21, s22 >= 0, s22 <= s20 + s21, s23 >= 0, s23 <= 1 + s19 + s22, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s24 + 34*x_1' + 12*x_1'^2 }-> s25 :|: s24 >= 0, s24 <= 6 * x_1', s25 >= 0, s25 <= 1 + s24 + 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s26 + s28 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s29 :|: s26 >= 0, s26 <= 6 * x_1'', s27 >= 0, s27 <= 6 * x_2', s28 >= 0, s28 <= s26 + s27, s29 >= 0, s29 <= s28 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s30 + s32 + 34*x_1'' + 12*x_1''^2 + 34*x_15 + 12*x_15^2 + 34*x_2' + 12*x_2'^2 }-> s34 :|: s30 >= 0, s30 <= 6 * x_1'', s31 >= 0, s31 <= 6 * x_2', s32 >= 0, s32 <= s30 + s31, s33 >= 0, s33 <= 6 * x_15, s34 >= 0, s34 <= s32 + (1 + s33), x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s35 + s37 + s38 + 34*x_1'' + 12*x_1''^2 + 34*x_16 + 12*x_16^2 + 34*x_2' + 12*x_2'^2 + 34*x_22 + 12*x_22^2 }-> s41 :|: s35 >= 0, s35 <= 6 * x_1'', s36 >= 0, s36 <= 6 * x_2', s37 >= 0, s37 <= s35 + s36, s38 >= 0, s38 <= 6 * x_16, s39 >= 0, s39 <= 6 * x_22, s40 >= 0, s40 <= s38 + s39, s41 >= 0, s41 <= s37 + s40, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s42 + s44 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s45 :|: s42 >= 0, s42 <= 6 * x_1'', s43 >= 0, s43 <= 6 * x_2', s44 >= 0, s44 <= s42 + s43, s45 >= 0, s45 <= s44 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 34*x_17 + 12*x_17^2 }-> s47 :|: s46 >= 0, s46 <= 6 * x_17, s47 >= 0, s47 <= 0 + (1 + s46), x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s48 + 34*x_18 + 12*x_18^2 + 34*x_23 + 12*x_23^2 }-> s51 :|: s48 >= 0, s48 <= 6 * x_18, s49 >= 0, s49 <= 6 * x_23, s50 >= 0, s50 <= s48 + s49, s51 >= 0, s51 <= 0 + s50, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ -18 + -14*z + 12*z^2 }-> s9 :|: s8 >= 0, s8 <= 6 * (z - 2), s9 >= 0, s9 <= 0 + (1 + s8), z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -21 + 10*z + 12*z^2 }-> 1 + s7 :|: s7 >= 0, s7 <= 6 * (z - 1), z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s53 :|: s52 >= 0, s52 <= 6 * (z' - 1), s53 >= 0, s53 <= 0 + (1 + s52), z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s54 + 34*x_112 + 12*x_112^2 + 34*x_25 + 12*x_25^2 }-> s57 :|: s54 >= 0, s54 <= 6 * x_112, s55 >= 0, s55 <= 6 * x_25, s56 >= 0, s56 <= s54 + s55, s57 >= 0, s57 <= 0 + s56, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ -19 + s58 + 10*z + 12*z^2 }-> s59 :|: s58 >= 0, s58 <= 6 * (z - 1), s59 >= 0, s59 <= 1 + s58 + 0, z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ -40 + s60 + 10*z + 12*z^2 + 10*z' + 12*z'^2 }-> s62 :|: s60 >= 0, s60 <= 6 * (z - 1), s61 >= 0, s61 <= 6 * (z' - 1), s62 >= 0, s62 <= 1 + s60 + (1 + s61), z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ -16 + s63 + s64 + 34*x_114 + 12*x_114^2 + 34*x_26 + 12*x_26^2 + 10*z + 12*z^2 }-> s67 :|: s63 >= 0, s63 <= 6 * (z - 1), s64 >= 0, s64 <= 6 * x_114, s65 >= 0, s65 <= 6 * x_26, s66 >= 0, s66 <= s64 + s65, s67 >= 0, s67 <= 1 + s63 + s66, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ -19 + s68 + 10*z + 12*z^2 }-> s69 :|: s68 >= 0, s68 <= 6 * (z - 1), s69 >= 0, s69 <= 1 + s68 + 0, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 4 + s70 + s72 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s73 :|: s70 >= 0, s70 <= 6 * x_110, s71 >= 0, s71 <= 6 * x_24, s72 >= 0, s72 <= s70 + s71, s73 >= 0, s73 <= s72 + 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ -17 + s74 + s76 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 + 10*z' + 12*z'^2 }-> s78 :|: s74 >= 0, s74 <= 6 * x_110, s75 >= 0, s75 <= 6 * x_24, s76 >= 0, s76 <= s74 + s75, s77 >= 0, s77 <= 6 * (z' - 1), s78 >= 0, s78 <= s76 + (1 + s77), z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 7 + s79 + s81 + s82 + 34*x_110 + 12*x_110^2 + 34*x_116 + 12*x_116^2 + 34*x_24 + 12*x_24^2 + 34*x_27 + 12*x_27^2 }-> s85 :|: s79 >= 0, s79 <= 6 * x_110, s80 >= 0, s80 <= 6 * x_24, s81 >= 0, s81 <= s79 + s80, s82 >= 0, s82 <= 6 * x_116, s83 >= 0, s83 <= 6 * x_27, s84 >= 0, s84 <= s82 + s83, s85 >= 0, s85 <= s81 + s84, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 4 + s86 + s88 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s89 :|: s86 >= 0, s86 <= 6 * x_110, s87 >= 0, s87 <= 6 * x_24, s88 >= 0, s88 <= s86 + s87, s89 >= 0, s89 <= s88 + 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s91 :|: s90 >= 0, s90 <= 6 * (z' - 1), s91 >= 0, s91 <= 0 + (1 + s90), z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s92 + 34*x_118 + 12*x_118^2 + 34*x_28 + 12*x_28^2 }-> s95 :|: s92 >= 0, s92 <= 6 * x_118, s93 >= 0, s93 <= 6 * x_28, s94 >= 0, s94 <= s92 + s93, s95 >= 0, s95 <= 0 + s94, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 34*z + 12*z^2 }-> 1 + s96 :|: s96 >= 0, s96 <= 6 * z, z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_+}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [6*z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 4 + s10 + 34*x_12 + 12*x_12^2 + 34*x_2'' + 12*x_2''^2 }-> s13 :|: s10 >= 0, s10 <= 6 * x_12, s11 >= 0, s11 <= 6 * x_2'', s12 >= 0, s12 <= s10 + s11, s13 >= 0, s13 <= 0 + s12, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ -17 + s14 + -14*z + 12*z^2 }-> s15 :|: s14 >= 0, s14 <= 6 * (z - 2), s15 >= 0, s15 <= 1 + s14 + 0, z - 2 >= 0
   encArg(z) -{ 4 + s16 + 34*x_1' + 12*x_1'^2 + 34*x_13 + 12*x_13^2 }-> s18 :|: s16 >= 0, s16 <= 6 * x_1', s17 >= 0, s17 <= 6 * x_13, s18 >= 0, s18 <= 1 + s16 + (1 + s17), z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 6 + s19 + s20 + 34*x_1' + 12*x_1'^2 + 34*x_14 + 12*x_14^2 + 34*x_21 + 12*x_21^2 }-> s23 :|: s19 >= 0, s19 <= 6 * x_1', s20 >= 0, s20 <= 6 * x_14, s21 >= 0, s21 <= 6 * x_21, s22 >= 0, s22 <= s20 + s21, s23 >= 0, s23 <= 1 + s19 + s22, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s24 + 34*x_1' + 12*x_1'^2 }-> s25 :|: s24 >= 0, s24 <= 6 * x_1', s25 >= 0, s25 <= 1 + s24 + 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s26 + s28 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s29 :|: s26 >= 0, s26 <= 6 * x_1'', s27 >= 0, s27 <= 6 * x_2', s28 >= 0, s28 <= s26 + s27, s29 >= 0, s29 <= s28 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s30 + s32 + 34*x_1'' + 12*x_1''^2 + 34*x_15 + 12*x_15^2 + 34*x_2' + 12*x_2'^2 }-> s34 :|: s30 >= 0, s30 <= 6 * x_1'', s31 >= 0, s31 <= 6 * x_2', s32 >= 0, s32 <= s30 + s31, s33 >= 0, s33 <= 6 * x_15, s34 >= 0, s34 <= s32 + (1 + s33), x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s35 + s37 + s38 + 34*x_1'' + 12*x_1''^2 + 34*x_16 + 12*x_16^2 + 34*x_2' + 12*x_2'^2 + 34*x_22 + 12*x_22^2 }-> s41 :|: s35 >= 0, s35 <= 6 * x_1'', s36 >= 0, s36 <= 6 * x_2', s37 >= 0, s37 <= s35 + s36, s38 >= 0, s38 <= 6 * x_16, s39 >= 0, s39 <= 6 * x_22, s40 >= 0, s40 <= s38 + s39, s41 >= 0, s41 <= s37 + s40, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s42 + s44 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s45 :|: s42 >= 0, s42 <= 6 * x_1'', s43 >= 0, s43 <= 6 * x_2', s44 >= 0, s44 <= s42 + s43, s45 >= 0, s45 <= s44 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 34*x_17 + 12*x_17^2 }-> s47 :|: s46 >= 0, s46 <= 6 * x_17, s47 >= 0, s47 <= 0 + (1 + s46), x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s48 + 34*x_18 + 12*x_18^2 + 34*x_23 + 12*x_23^2 }-> s51 :|: s48 >= 0, s48 <= 6 * x_18, s49 >= 0, s49 <= 6 * x_23, s50 >= 0, s50 <= s48 + s49, s51 >= 0, s51 <= 0 + s50, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ -18 + -14*z + 12*z^2 }-> s9 :|: s8 >= 0, s8 <= 6 * (z - 2), s9 >= 0, s9 <= 0 + (1 + s8), z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -21 + 10*z + 12*z^2 }-> 1 + s7 :|: s7 >= 0, s7 <= 6 * (z - 1), z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s53 :|: s52 >= 0, s52 <= 6 * (z' - 1), s53 >= 0, s53 <= 0 + (1 + s52), z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s54 + 34*x_112 + 12*x_112^2 + 34*x_25 + 12*x_25^2 }-> s57 :|: s54 >= 0, s54 <= 6 * x_112, s55 >= 0, s55 <= 6 * x_25, s56 >= 0, s56 <= s54 + s55, s57 >= 0, s57 <= 0 + s56, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ -19 + s58 + 10*z + 12*z^2 }-> s59 :|: s58 >= 0, s58 <= 6 * (z - 1), s59 >= 0, s59 <= 1 + s58 + 0, z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ -40 + s60 + 10*z + 12*z^2 + 10*z' + 12*z'^2 }-> s62 :|: s60 >= 0, s60 <= 6 * (z - 1), s61 >= 0, s61 <= 6 * (z' - 1), s62 >= 0, s62 <= 1 + s60 + (1 + s61), z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ -16 + s63 + s64 + 34*x_114 + 12*x_114^2 + 34*x_26 + 12*x_26^2 + 10*z + 12*z^2 }-> s67 :|: s63 >= 0, s63 <= 6 * (z - 1), s64 >= 0, s64 <= 6 * x_114, s65 >= 0, s65 <= 6 * x_26, s66 >= 0, s66 <= s64 + s65, s67 >= 0, s67 <= 1 + s63 + s66, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ -19 + s68 + 10*z + 12*z^2 }-> s69 :|: s68 >= 0, s68 <= 6 * (z - 1), s69 >= 0, s69 <= 1 + s68 + 0, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 4 + s70 + s72 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s73 :|: s70 >= 0, s70 <= 6 * x_110, s71 >= 0, s71 <= 6 * x_24, s72 >= 0, s72 <= s70 + s71, s73 >= 0, s73 <= s72 + 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ -17 + s74 + s76 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 + 10*z' + 12*z'^2 }-> s78 :|: s74 >= 0, s74 <= 6 * x_110, s75 >= 0, s75 <= 6 * x_24, s76 >= 0, s76 <= s74 + s75, s77 >= 0, s77 <= 6 * (z' - 1), s78 >= 0, s78 <= s76 + (1 + s77), z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 7 + s79 + s81 + s82 + 34*x_110 + 12*x_110^2 + 34*x_116 + 12*x_116^2 + 34*x_24 + 12*x_24^2 + 34*x_27 + 12*x_27^2 }-> s85 :|: s79 >= 0, s79 <= 6 * x_110, s80 >= 0, s80 <= 6 * x_24, s81 >= 0, s81 <= s79 + s80, s82 >= 0, s82 <= 6 * x_116, s83 >= 0, s83 <= 6 * x_27, s84 >= 0, s84 <= s82 + s83, s85 >= 0, s85 <= s81 + s84, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 4 + s86 + s88 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s89 :|: s86 >= 0, s86 <= 6 * x_110, s87 >= 0, s87 <= 6 * x_24, s88 >= 0, s88 <= s86 + s87, s89 >= 0, s89 <= s88 + 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s91 :|: s90 >= 0, s90 <= 6 * (z' - 1), s91 >= 0, s91 <= 0 + (1 + s90), z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s92 + 34*x_118 + 12*x_118^2 + 34*x_28 + 12*x_28^2 }-> s95 :|: s92 >= 0, s92 <= 6 * x_118, s93 >= 0, s93 <= 6 * x_28, s94 >= 0, s94 <= s92 + s93, s95 >= 0, s95 <= 0 + s94, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 34*z + 12*z^2 }-> 1 + s96 :|: s96 >= 0, s96 <= 6 * z, z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_+}, {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [6*z]
  encode_+: runtime: ?, size: O(n^1) [6*z + 6*z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_+
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 27 + 402*z + 144*z^2 + 336*z' + 144*z'^2

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

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

   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 4 + s10 + 34*x_12 + 12*x_12^2 + 34*x_2'' + 12*x_2''^2 }-> s13 :|: s10 >= 0, s10 <= 6 * x_12, s11 >= 0, s11 <= 6 * x_2'', s12 >= 0, s12 <= s10 + s11, s13 >= 0, s13 <= 0 + s12, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ -17 + s14 + -14*z + 12*z^2 }-> s15 :|: s14 >= 0, s14 <= 6 * (z - 2), s15 >= 0, s15 <= 1 + s14 + 0, z - 2 >= 0
   encArg(z) -{ 4 + s16 + 34*x_1' + 12*x_1'^2 + 34*x_13 + 12*x_13^2 }-> s18 :|: s16 >= 0, s16 <= 6 * x_1', s17 >= 0, s17 <= 6 * x_13, s18 >= 0, s18 <= 1 + s16 + (1 + s17), z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 6 + s19 + s20 + 34*x_1' + 12*x_1'^2 + 34*x_14 + 12*x_14^2 + 34*x_21 + 12*x_21^2 }-> s23 :|: s19 >= 0, s19 <= 6 * x_1', s20 >= 0, s20 <= 6 * x_14, s21 >= 0, s21 <= 6 * x_21, s22 >= 0, s22 <= s20 + s21, s23 >= 0, s23 <= 1 + s19 + s22, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s24 + 34*x_1' + 12*x_1'^2 }-> s25 :|: s24 >= 0, s24 <= 6 * x_1', s25 >= 0, s25 <= 1 + s24 + 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s26 + s28 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s29 :|: s26 >= 0, s26 <= 6 * x_1'', s27 >= 0, s27 <= 6 * x_2', s28 >= 0, s28 <= s26 + s27, s29 >= 0, s29 <= s28 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s30 + s32 + 34*x_1'' + 12*x_1''^2 + 34*x_15 + 12*x_15^2 + 34*x_2' + 12*x_2'^2 }-> s34 :|: s30 >= 0, s30 <= 6 * x_1'', s31 >= 0, s31 <= 6 * x_2', s32 >= 0, s32 <= s30 + s31, s33 >= 0, s33 <= 6 * x_15, s34 >= 0, s34 <= s32 + (1 + s33), x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s35 + s37 + s38 + 34*x_1'' + 12*x_1''^2 + 34*x_16 + 12*x_16^2 + 34*x_2' + 12*x_2'^2 + 34*x_22 + 12*x_22^2 }-> s41 :|: s35 >= 0, s35 <= 6 * x_1'', s36 >= 0, s36 <= 6 * x_2', s37 >= 0, s37 <= s35 + s36, s38 >= 0, s38 <= 6 * x_16, s39 >= 0, s39 <= 6 * x_22, s40 >= 0, s40 <= s38 + s39, s41 >= 0, s41 <= s37 + s40, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s42 + s44 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s45 :|: s42 >= 0, s42 <= 6 * x_1'', s43 >= 0, s43 <= 6 * x_2', s44 >= 0, s44 <= s42 + s43, s45 >= 0, s45 <= s44 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 34*x_17 + 12*x_17^2 }-> s47 :|: s46 >= 0, s46 <= 6 * x_17, s47 >= 0, s47 <= 0 + (1 + s46), x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s48 + 34*x_18 + 12*x_18^2 + 34*x_23 + 12*x_23^2 }-> s51 :|: s48 >= 0, s48 <= 6 * x_18, s49 >= 0, s49 <= 6 * x_23, s50 >= 0, s50 <= s48 + s49, s51 >= 0, s51 <= 0 + s50, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ -18 + -14*z + 12*z^2 }-> s9 :|: s8 >= 0, s8 <= 6 * (z - 2), s9 >= 0, s9 <= 0 + (1 + s8), z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -21 + 10*z + 12*z^2 }-> 1 + s7 :|: s7 >= 0, s7 <= 6 * (z - 1), z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s53 :|: s52 >= 0, s52 <= 6 * (z' - 1), s53 >= 0, s53 <= 0 + (1 + s52), z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s54 + 34*x_112 + 12*x_112^2 + 34*x_25 + 12*x_25^2 }-> s57 :|: s54 >= 0, s54 <= 6 * x_112, s55 >= 0, s55 <= 6 * x_25, s56 >= 0, s56 <= s54 + s55, s57 >= 0, s57 <= 0 + s56, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ -19 + s58 + 10*z + 12*z^2 }-> s59 :|: s58 >= 0, s58 <= 6 * (z - 1), s59 >= 0, s59 <= 1 + s58 + 0, z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ -40 + s60 + 10*z + 12*z^2 + 10*z' + 12*z'^2 }-> s62 :|: s60 >= 0, s60 <= 6 * (z - 1), s61 >= 0, s61 <= 6 * (z' - 1), s62 >= 0, s62 <= 1 + s60 + (1 + s61), z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ -16 + s63 + s64 + 34*x_114 + 12*x_114^2 + 34*x_26 + 12*x_26^2 + 10*z + 12*z^2 }-> s67 :|: s63 >= 0, s63 <= 6 * (z - 1), s64 >= 0, s64 <= 6 * x_114, s65 >= 0, s65 <= 6 * x_26, s66 >= 0, s66 <= s64 + s65, s67 >= 0, s67 <= 1 + s63 + s66, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ -19 + s68 + 10*z + 12*z^2 }-> s69 :|: s68 >= 0, s68 <= 6 * (z - 1), s69 >= 0, s69 <= 1 + s68 + 0, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 4 + s70 + s72 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s73 :|: s70 >= 0, s70 <= 6 * x_110, s71 >= 0, s71 <= 6 * x_24, s72 >= 0, s72 <= s70 + s71, s73 >= 0, s73 <= s72 + 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ -17 + s74 + s76 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 + 10*z' + 12*z'^2 }-> s78 :|: s74 >= 0, s74 <= 6 * x_110, s75 >= 0, s75 <= 6 * x_24, s76 >= 0, s76 <= s74 + s75, s77 >= 0, s77 <= 6 * (z' - 1), s78 >= 0, s78 <= s76 + (1 + s77), z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 7 + s79 + s81 + s82 + 34*x_110 + 12*x_110^2 + 34*x_116 + 12*x_116^2 + 34*x_24 + 12*x_24^2 + 34*x_27 + 12*x_27^2 }-> s85 :|: s79 >= 0, s79 <= 6 * x_110, s80 >= 0, s80 <= 6 * x_24, s81 >= 0, s81 <= s79 + s80, s82 >= 0, s82 <= 6 * x_116, s83 >= 0, s83 <= 6 * x_27, s84 >= 0, s84 <= s82 + s83, s85 >= 0, s85 <= s81 + s84, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 4 + s86 + s88 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s89 :|: s86 >= 0, s86 <= 6 * x_110, s87 >= 0, s87 <= 6 * x_24, s88 >= 0, s88 <= s86 + s87, s89 >= 0, s89 <= s88 + 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s91 :|: s90 >= 0, s90 <= 6 * (z' - 1), s91 >= 0, s91 <= 0 + (1 + s90), z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s92 + 34*x_118 + 12*x_118^2 + 34*x_28 + 12*x_28^2 }-> s95 :|: s92 >= 0, s92 <= 6 * x_118, s93 >= 0, s93 <= 6 * x_28, s94 >= 0, s94 <= s92 + s93, s95 >= 0, s95 <= 0 + s94, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 34*z + 12*z^2 }-> 1 + s96 :|: s96 >= 0, s96 <= 6 * z, z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [6*z]
  encode_+: runtime: O(n^2) [27 + 402*z + 144*z^2 + 336*z' + 144*z'^2], size: O(n^1) [6*z + 6*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) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 4 + s10 + 34*x_12 + 12*x_12^2 + 34*x_2'' + 12*x_2''^2 }-> s13 :|: s10 >= 0, s10 <= 6 * x_12, s11 >= 0, s11 <= 6 * x_2'', s12 >= 0, s12 <= s10 + s11, s13 >= 0, s13 <= 0 + s12, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ -17 + s14 + -14*z + 12*z^2 }-> s15 :|: s14 >= 0, s14 <= 6 * (z - 2), s15 >= 0, s15 <= 1 + s14 + 0, z - 2 >= 0
   encArg(z) -{ 4 + s16 + 34*x_1' + 12*x_1'^2 + 34*x_13 + 12*x_13^2 }-> s18 :|: s16 >= 0, s16 <= 6 * x_1', s17 >= 0, s17 <= 6 * x_13, s18 >= 0, s18 <= 1 + s16 + (1 + s17), z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 6 + s19 + s20 + 34*x_1' + 12*x_1'^2 + 34*x_14 + 12*x_14^2 + 34*x_21 + 12*x_21^2 }-> s23 :|: s19 >= 0, s19 <= 6 * x_1', s20 >= 0, s20 <= 6 * x_14, s21 >= 0, s21 <= 6 * x_21, s22 >= 0, s22 <= s20 + s21, s23 >= 0, s23 <= 1 + s19 + s22, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s24 + 34*x_1' + 12*x_1'^2 }-> s25 :|: s24 >= 0, s24 <= 6 * x_1', s25 >= 0, s25 <= 1 + s24 + 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s26 + s28 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s29 :|: s26 >= 0, s26 <= 6 * x_1'', s27 >= 0, s27 <= 6 * x_2', s28 >= 0, s28 <= s26 + s27, s29 >= 0, s29 <= s28 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s30 + s32 + 34*x_1'' + 12*x_1''^2 + 34*x_15 + 12*x_15^2 + 34*x_2' + 12*x_2'^2 }-> s34 :|: s30 >= 0, s30 <= 6 * x_1'', s31 >= 0, s31 <= 6 * x_2', s32 >= 0, s32 <= s30 + s31, s33 >= 0, s33 <= 6 * x_15, s34 >= 0, s34 <= s32 + (1 + s33), x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s35 + s37 + s38 + 34*x_1'' + 12*x_1''^2 + 34*x_16 + 12*x_16^2 + 34*x_2' + 12*x_2'^2 + 34*x_22 + 12*x_22^2 }-> s41 :|: s35 >= 0, s35 <= 6 * x_1'', s36 >= 0, s36 <= 6 * x_2', s37 >= 0, s37 <= s35 + s36, s38 >= 0, s38 <= 6 * x_16, s39 >= 0, s39 <= 6 * x_22, s40 >= 0, s40 <= s38 + s39, s41 >= 0, s41 <= s37 + s40, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s42 + s44 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s45 :|: s42 >= 0, s42 <= 6 * x_1'', s43 >= 0, s43 <= 6 * x_2', s44 >= 0, s44 <= s42 + s43, s45 >= 0, s45 <= s44 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 34*x_17 + 12*x_17^2 }-> s47 :|: s46 >= 0, s46 <= 6 * x_17, s47 >= 0, s47 <= 0 + (1 + s46), x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s48 + 34*x_18 + 12*x_18^2 + 34*x_23 + 12*x_23^2 }-> s51 :|: s48 >= 0, s48 <= 6 * x_18, s49 >= 0, s49 <= 6 * x_23, s50 >= 0, s50 <= s48 + s49, s51 >= 0, s51 <= 0 + s50, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ -18 + -14*z + 12*z^2 }-> s9 :|: s8 >= 0, s8 <= 6 * (z - 2), s9 >= 0, s9 <= 0 + (1 + s8), z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -21 + 10*z + 12*z^2 }-> 1 + s7 :|: s7 >= 0, s7 <= 6 * (z - 1), z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s53 :|: s52 >= 0, s52 <= 6 * (z' - 1), s53 >= 0, s53 <= 0 + (1 + s52), z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s54 + 34*x_112 + 12*x_112^2 + 34*x_25 + 12*x_25^2 }-> s57 :|: s54 >= 0, s54 <= 6 * x_112, s55 >= 0, s55 <= 6 * x_25, s56 >= 0, s56 <= s54 + s55, s57 >= 0, s57 <= 0 + s56, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ -19 + s58 + 10*z + 12*z^2 }-> s59 :|: s58 >= 0, s58 <= 6 * (z - 1), s59 >= 0, s59 <= 1 + s58 + 0, z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ -40 + s60 + 10*z + 12*z^2 + 10*z' + 12*z'^2 }-> s62 :|: s60 >= 0, s60 <= 6 * (z - 1), s61 >= 0, s61 <= 6 * (z' - 1), s62 >= 0, s62 <= 1 + s60 + (1 + s61), z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ -16 + s63 + s64 + 34*x_114 + 12*x_114^2 + 34*x_26 + 12*x_26^2 + 10*z + 12*z^2 }-> s67 :|: s63 >= 0, s63 <= 6 * (z - 1), s64 >= 0, s64 <= 6 * x_114, s65 >= 0, s65 <= 6 * x_26, s66 >= 0, s66 <= s64 + s65, s67 >= 0, s67 <= 1 + s63 + s66, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ -19 + s68 + 10*z + 12*z^2 }-> s69 :|: s68 >= 0, s68 <= 6 * (z - 1), s69 >= 0, s69 <= 1 + s68 + 0, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 4 + s70 + s72 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s73 :|: s70 >= 0, s70 <= 6 * x_110, s71 >= 0, s71 <= 6 * x_24, s72 >= 0, s72 <= s70 + s71, s73 >= 0, s73 <= s72 + 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ -17 + s74 + s76 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 + 10*z' + 12*z'^2 }-> s78 :|: s74 >= 0, s74 <= 6 * x_110, s75 >= 0, s75 <= 6 * x_24, s76 >= 0, s76 <= s74 + s75, s77 >= 0, s77 <= 6 * (z' - 1), s78 >= 0, s78 <= s76 + (1 + s77), z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 7 + s79 + s81 + s82 + 34*x_110 + 12*x_110^2 + 34*x_116 + 12*x_116^2 + 34*x_24 + 12*x_24^2 + 34*x_27 + 12*x_27^2 }-> s85 :|: s79 >= 0, s79 <= 6 * x_110, s80 >= 0, s80 <= 6 * x_24, s81 >= 0, s81 <= s79 + s80, s82 >= 0, s82 <= 6 * x_116, s83 >= 0, s83 <= 6 * x_27, s84 >= 0, s84 <= s82 + s83, s85 >= 0, s85 <= s81 + s84, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 4 + s86 + s88 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s89 :|: s86 >= 0, s86 <= 6 * x_110, s87 >= 0, s87 <= 6 * x_24, s88 >= 0, s88 <= s86 + s87, s89 >= 0, s89 <= s88 + 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s91 :|: s90 >= 0, s90 <= 6 * (z' - 1), s91 >= 0, s91 <= 0 + (1 + s90), z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s92 + 34*x_118 + 12*x_118^2 + 34*x_28 + 12*x_28^2 }-> s95 :|: s92 >= 0, s92 <= 6 * x_118, s93 >= 0, s93 <= 6 * x_28, s94 >= 0, s94 <= s92 + s93, s95 >= 0, s95 <= 0 + s94, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 34*z + 12*z^2 }-> 1 + s96 :|: s96 >= 0, s96 <= 6 * z, z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [6*z]
  encode_+: runtime: O(n^2) [27 + 402*z + 144*z^2 + 336*z' + 144*z'^2], size: O(n^1) [6*z + 6*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: 1 + 6*z

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

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

   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 4 + s10 + 34*x_12 + 12*x_12^2 + 34*x_2'' + 12*x_2''^2 }-> s13 :|: s10 >= 0, s10 <= 6 * x_12, s11 >= 0, s11 <= 6 * x_2'', s12 >= 0, s12 <= s10 + s11, s13 >= 0, s13 <= 0 + s12, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ -17 + s14 + -14*z + 12*z^2 }-> s15 :|: s14 >= 0, s14 <= 6 * (z - 2), s15 >= 0, s15 <= 1 + s14 + 0, z - 2 >= 0
   encArg(z) -{ 4 + s16 + 34*x_1' + 12*x_1'^2 + 34*x_13 + 12*x_13^2 }-> s18 :|: s16 >= 0, s16 <= 6 * x_1', s17 >= 0, s17 <= 6 * x_13, s18 >= 0, s18 <= 1 + s16 + (1 + s17), z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 6 + s19 + s20 + 34*x_1' + 12*x_1'^2 + 34*x_14 + 12*x_14^2 + 34*x_21 + 12*x_21^2 }-> s23 :|: s19 >= 0, s19 <= 6 * x_1', s20 >= 0, s20 <= 6 * x_14, s21 >= 0, s21 <= 6 * x_21, s22 >= 0, s22 <= s20 + s21, s23 >= 0, s23 <= 1 + s19 + s22, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s24 + 34*x_1' + 12*x_1'^2 }-> s25 :|: s24 >= 0, s24 <= 6 * x_1', s25 >= 0, s25 <= 1 + s24 + 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s26 + s28 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s29 :|: s26 >= 0, s26 <= 6 * x_1'', s27 >= 0, s27 <= 6 * x_2', s28 >= 0, s28 <= s26 + s27, s29 >= 0, s29 <= s28 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s30 + s32 + 34*x_1'' + 12*x_1''^2 + 34*x_15 + 12*x_15^2 + 34*x_2' + 12*x_2'^2 }-> s34 :|: s30 >= 0, s30 <= 6 * x_1'', s31 >= 0, s31 <= 6 * x_2', s32 >= 0, s32 <= s30 + s31, s33 >= 0, s33 <= 6 * x_15, s34 >= 0, s34 <= s32 + (1 + s33), x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s35 + s37 + s38 + 34*x_1'' + 12*x_1''^2 + 34*x_16 + 12*x_16^2 + 34*x_2' + 12*x_2'^2 + 34*x_22 + 12*x_22^2 }-> s41 :|: s35 >= 0, s35 <= 6 * x_1'', s36 >= 0, s36 <= 6 * x_2', s37 >= 0, s37 <= s35 + s36, s38 >= 0, s38 <= 6 * x_16, s39 >= 0, s39 <= 6 * x_22, s40 >= 0, s40 <= s38 + s39, s41 >= 0, s41 <= s37 + s40, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s42 + s44 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s45 :|: s42 >= 0, s42 <= 6 * x_1'', s43 >= 0, s43 <= 6 * x_2', s44 >= 0, s44 <= s42 + s43, s45 >= 0, s45 <= s44 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 34*x_17 + 12*x_17^2 }-> s47 :|: s46 >= 0, s46 <= 6 * x_17, s47 >= 0, s47 <= 0 + (1 + s46), x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s48 + 34*x_18 + 12*x_18^2 + 34*x_23 + 12*x_23^2 }-> s51 :|: s48 >= 0, s48 <= 6 * x_18, s49 >= 0, s49 <= 6 * x_23, s50 >= 0, s50 <= s48 + s49, s51 >= 0, s51 <= 0 + s50, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ -18 + -14*z + 12*z^2 }-> s9 :|: s8 >= 0, s8 <= 6 * (z - 2), s9 >= 0, s9 <= 0 + (1 + s8), z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -21 + 10*z + 12*z^2 }-> 1 + s7 :|: s7 >= 0, s7 <= 6 * (z - 1), z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s53 :|: s52 >= 0, s52 <= 6 * (z' - 1), s53 >= 0, s53 <= 0 + (1 + s52), z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s54 + 34*x_112 + 12*x_112^2 + 34*x_25 + 12*x_25^2 }-> s57 :|: s54 >= 0, s54 <= 6 * x_112, s55 >= 0, s55 <= 6 * x_25, s56 >= 0, s56 <= s54 + s55, s57 >= 0, s57 <= 0 + s56, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ -19 + s58 + 10*z + 12*z^2 }-> s59 :|: s58 >= 0, s58 <= 6 * (z - 1), s59 >= 0, s59 <= 1 + s58 + 0, z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ -40 + s60 + 10*z + 12*z^2 + 10*z' + 12*z'^2 }-> s62 :|: s60 >= 0, s60 <= 6 * (z - 1), s61 >= 0, s61 <= 6 * (z' - 1), s62 >= 0, s62 <= 1 + s60 + (1 + s61), z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ -16 + s63 + s64 + 34*x_114 + 12*x_114^2 + 34*x_26 + 12*x_26^2 + 10*z + 12*z^2 }-> s67 :|: s63 >= 0, s63 <= 6 * (z - 1), s64 >= 0, s64 <= 6 * x_114, s65 >= 0, s65 <= 6 * x_26, s66 >= 0, s66 <= s64 + s65, s67 >= 0, s67 <= 1 + s63 + s66, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ -19 + s68 + 10*z + 12*z^2 }-> s69 :|: s68 >= 0, s68 <= 6 * (z - 1), s69 >= 0, s69 <= 1 + s68 + 0, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 4 + s70 + s72 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s73 :|: s70 >= 0, s70 <= 6 * x_110, s71 >= 0, s71 <= 6 * x_24, s72 >= 0, s72 <= s70 + s71, s73 >= 0, s73 <= s72 + 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ -17 + s74 + s76 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 + 10*z' + 12*z'^2 }-> s78 :|: s74 >= 0, s74 <= 6 * x_110, s75 >= 0, s75 <= 6 * x_24, s76 >= 0, s76 <= s74 + s75, s77 >= 0, s77 <= 6 * (z' - 1), s78 >= 0, s78 <= s76 + (1 + s77), z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 7 + s79 + s81 + s82 + 34*x_110 + 12*x_110^2 + 34*x_116 + 12*x_116^2 + 34*x_24 + 12*x_24^2 + 34*x_27 + 12*x_27^2 }-> s85 :|: s79 >= 0, s79 <= 6 * x_110, s80 >= 0, s80 <= 6 * x_24, s81 >= 0, s81 <= s79 + s80, s82 >= 0, s82 <= 6 * x_116, s83 >= 0, s83 <= 6 * x_27, s84 >= 0, s84 <= s82 + s83, s85 >= 0, s85 <= s81 + s84, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 4 + s86 + s88 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s89 :|: s86 >= 0, s86 <= 6 * x_110, s87 >= 0, s87 <= 6 * x_24, s88 >= 0, s88 <= s86 + s87, s89 >= 0, s89 <= s88 + 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s91 :|: s90 >= 0, s90 <= 6 * (z' - 1), s91 >= 0, s91 <= 0 + (1 + s90), z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s92 + 34*x_118 + 12*x_118^2 + 34*x_28 + 12*x_28^2 }-> s95 :|: s92 >= 0, s92 <= 6 * x_118, s93 >= 0, s93 <= 6 * x_28, s94 >= 0, s94 <= s92 + s93, s95 >= 0, s95 <= 0 + s94, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 34*z + 12*z^2 }-> 1 + s96 :|: s96 >= 0, s96 <= 6 * z, z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [6*z]
  encode_+: runtime: O(n^2) [27 + 402*z + 144*z^2 + 336*z' + 144*z'^2], size: O(n^1) [6*z + 6*z']
  encode_s: runtime: ?, size: O(n^1) [1 + 6*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^2) with polynomial bound: 1 + 34*z + 12*z^2

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

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

   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 0, z - 1 >= 0
   encArg(z) -{ 4 + s10 + 34*x_12 + 12*x_12^2 + 34*x_2'' + 12*x_2''^2 }-> s13 :|: s10 >= 0, s10 <= 6 * x_12, s11 >= 0, s11 <= 6 * x_2'', s12 >= 0, s12 <= s10 + s11, s13 >= 0, s13 <= 0 + s12, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ -17 + s14 + -14*z + 12*z^2 }-> s15 :|: s14 >= 0, s14 <= 6 * (z - 2), s15 >= 0, s15 <= 1 + s14 + 0, z - 2 >= 0
   encArg(z) -{ 4 + s16 + 34*x_1' + 12*x_1'^2 + 34*x_13 + 12*x_13^2 }-> s18 :|: s16 >= 0, s16 <= 6 * x_1', s17 >= 0, s17 <= 6 * x_13, s18 >= 0, s18 <= 1 + s16 + (1 + s17), z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 6 + s19 + s20 + 34*x_1' + 12*x_1'^2 + 34*x_14 + 12*x_14^2 + 34*x_21 + 12*x_21^2 }-> s23 :|: s19 >= 0, s19 <= 6 * x_1', s20 >= 0, s20 <= 6 * x_14, s21 >= 0, s21 <= 6 * x_21, s22 >= 0, s22 <= s20 + s21, s23 >= 0, s23 <= 1 + s19 + s22, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s24 + 34*x_1' + 12*x_1'^2 }-> s25 :|: s24 >= 0, s24 <= 6 * x_1', s25 >= 0, s25 <= 1 + s24 + 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s26 + s28 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s29 :|: s26 >= 0, s26 <= 6 * x_1'', s27 >= 0, s27 <= 6 * x_2', s28 >= 0, s28 <= s26 + s27, s29 >= 0, s29 <= s28 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s30 + s32 + 34*x_1'' + 12*x_1''^2 + 34*x_15 + 12*x_15^2 + 34*x_2' + 12*x_2'^2 }-> s34 :|: s30 >= 0, s30 <= 6 * x_1'', s31 >= 0, s31 <= 6 * x_2', s32 >= 0, s32 <= s30 + s31, s33 >= 0, s33 <= 6 * x_15, s34 >= 0, s34 <= s32 + (1 + s33), x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s35 + s37 + s38 + 34*x_1'' + 12*x_1''^2 + 34*x_16 + 12*x_16^2 + 34*x_2' + 12*x_2'^2 + 34*x_22 + 12*x_22^2 }-> s41 :|: s35 >= 0, s35 <= 6 * x_1'', s36 >= 0, s36 <= 6 * x_2', s37 >= 0, s37 <= s35 + s36, s38 >= 0, s38 <= 6 * x_16, s39 >= 0, s39 <= 6 * x_22, s40 >= 0, s40 <= s38 + s39, s41 >= 0, s41 <= s37 + s40, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s42 + s44 + 34*x_1'' + 12*x_1''^2 + 34*x_2' + 12*x_2'^2 }-> s45 :|: s42 >= 0, s42 <= 6 * x_1'', s43 >= 0, s43 <= 6 * x_2', s44 >= 0, s44 <= s42 + s43, s45 >= 0, s45 <= s44 + 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 34*x_17 + 12*x_17^2 }-> s47 :|: s46 >= 0, s46 <= 6 * x_17, s47 >= 0, s47 <= 0 + (1 + s46), x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s48 + 34*x_18 + 12*x_18^2 + 34*x_23 + 12*x_23^2 }-> s51 :|: s48 >= 0, s48 <= 6 * x_18, s49 >= 0, s49 <= 6 * x_23, s50 >= 0, s50 <= s48 + s49, s51 >= 0, s51 <= 0 + s50, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ -18 + -14*z + 12*z^2 }-> s9 :|: s8 >= 0, s8 <= 6 * (z - 2), s9 >= 0, s9 <= 0 + (1 + s8), z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -21 + 10*z + 12*z^2 }-> 1 + s7 :|: s7 >= 0, s7 <= 6 * (z - 1), z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 0, z = 0, z' = 0
   encode_+(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 0, z' >= 0, z = 0
   encode_+(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z' = 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s53 :|: s52 >= 0, s52 <= 6 * (z' - 1), s53 >= 0, s53 <= 0 + (1 + s52), z = 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s54 + 34*x_112 + 12*x_112^2 + 34*x_25 + 12*x_25^2 }-> s57 :|: s54 >= 0, s54 <= 6 * x_112, s55 >= 0, s55 <= 6 * x_25, s56 >= 0, s56 <= s54 + s55, s57 >= 0, s57 <= 0 + s56, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_+(z, z') -{ -19 + s58 + 10*z + 12*z^2 }-> s59 :|: s58 >= 0, s58 <= 6 * (z - 1), s59 >= 0, s59 <= 1 + s58 + 0, z' = 0, z - 1 >= 0
   encode_+(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 0 + 0, z >= 0, z' >= 0
   encode_+(z, z') -{ -40 + s60 + 10*z + 12*z^2 + 10*z' + 12*z'^2 }-> s62 :|: s60 >= 0, s60 <= 6 * (z - 1), s61 >= 0, s61 <= 6 * (z' - 1), s62 >= 0, s62 <= 1 + s60 + (1 + s61), z' - 1 >= 0, z - 1 >= 0
   encode_+(z, z') -{ -16 + s63 + s64 + 34*x_114 + 12*x_114^2 + 34*x_26 + 12*x_26^2 + 10*z + 12*z^2 }-> s67 :|: s63 >= 0, s63 <= 6 * (z - 1), s64 >= 0, s64 <= 6 * x_114, s65 >= 0, s65 <= 6 * x_26, s66 >= 0, s66 <= s64 + s65, s67 >= 0, s67 <= 1 + s63 + s66, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_+(z, z') -{ -19 + s68 + 10*z + 12*z^2 }-> s69 :|: s68 >= 0, s68 <= 6 * (z - 1), s69 >= 0, s69 <= 1 + s68 + 0, z' >= 0, z - 1 >= 0
   encode_+(z, z') -{ 4 + s70 + s72 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s73 :|: s70 >= 0, s70 <= 6 * x_110, s71 >= 0, s71 <= 6 * x_24, s72 >= 0, s72 <= s70 + s71, s73 >= 0, s73 <= s72 + 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_+(z, z') -{ -17 + s74 + s76 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 + 10*z' + 12*z'^2 }-> s78 :|: s74 >= 0, s74 <= 6 * x_110, s75 >= 0, s75 <= 6 * x_24, s76 >= 0, s76 <= s74 + s75, s77 >= 0, s77 <= 6 * (z' - 1), s78 >= 0, s78 <= s76 + (1 + s77), z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_+(z, z') -{ 7 + s79 + s81 + s82 + 34*x_110 + 12*x_110^2 + 34*x_116 + 12*x_116^2 + 34*x_24 + 12*x_24^2 + 34*x_27 + 12*x_27^2 }-> s85 :|: s79 >= 0, s79 <= 6 * x_110, s80 >= 0, s80 <= 6 * x_24, s81 >= 0, s81 <= s79 + s80, s82 >= 0, s82 <= 6 * x_116, s83 >= 0, s83 <= 6 * x_27, s84 >= 0, s84 <= s82 + s83, s85 >= 0, s85 <= s81 + s84, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_+(z, z') -{ 4 + s86 + s88 + 34*x_110 + 12*x_110^2 + 34*x_24 + 12*x_24^2 }-> s89 :|: s86 >= 0, s86 <= 6 * x_110, s87 >= 0, s87 <= 6 * x_24, s88 >= 0, s88 <= s86 + s87, s89 >= 0, s89 <= s88 + 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_+(z, z') -{ -20 + 10*z' + 12*z'^2 }-> s91 :|: s90 >= 0, s90 <= 6 * (z' - 1), s91 >= 0, s91 <= 0 + (1 + s90), z >= 0, z' - 1 >= 0
   encode_+(z, z') -{ 4 + s92 + 34*x_118 + 12*x_118^2 + 34*x_28 + 12*x_28^2 }-> s95 :|: s92 >= 0, s92 <= 6 * x_118, s93 >= 0, s93 <= 6 * x_28, s94 >= 0, s94 <= s92 + s93, s95 >= 0, s95 <= 0 + s94, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_+(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 34*z + 12*z^2 }-> 1 + s96 :|: s96 >= 0, s96 <= 6 * z, z >= 0
   plus(z, z') -{ 1 + z }-> s' :|: s' >= 0, s' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_0: runtime: O(1) [0], size: O(1) [0]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  encArg: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [6*z]
  encode_+: runtime: O(n^2) [27 + 402*z + 144*z^2 + 336*z' + 144*z'^2], size: O(n^1) [6*z + 6*z']
  encode_s: runtime: O(n^2) [1 + 34*z + 12*z^2], size: O(n^1) [1 + 6*z]

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(53) FinalProof (FINISHED)
Computed overall runtime complexity
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(54)
BOUNDS(1, n^2)

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

(56)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

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

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

Rewrite Strategy: FULL
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(57) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(58)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(s(x), y) -> +'(x, s(y))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
+' :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
0' :: 0':s:cons_+
s :: 0':s:cons_+ -> 0':s:cons_+
encArg :: 0':s:cons_+ -> 0':s:cons_+
cons_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_0 :: 0':s:cons_+
encode_s :: 0':s:cons_+ -> 0':s:cons_+
hole_0':s:cons_+1_3 :: 0':s:cons_+
gen_0':s:cons_+2_3 :: Nat -> 0':s:cons_+

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

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

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(60)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(s(x), y) -> +'(x, s(y))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))

Types:
+' :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
0' :: 0':s:cons_+
s :: 0':s:cons_+ -> 0':s:cons_+
encArg :: 0':s:cons_+ -> 0':s:cons_+
cons_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_0 :: 0':s:cons_+
encode_s :: 0':s:cons_+ -> 0':s:cons_+
hole_0':s:cons_+1_3 :: 0':s:cons_+
gen_0':s:cons_+2_3 :: Nat -> 0':s:cons_+


Generator Equations:
gen_0':s:cons_+2_3(0) <=> 0'
gen_0':s:cons_+2_3(+(x, 1)) <=> s(gen_0':s:cons_+2_3(x))


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

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

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(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:cons_+2_3(n4_3), gen_0':s:cons_+2_3(b)) -> gen_0':s:cons_+2_3(+(n4_3, b)), rt in Omega(1 + n4_3)

Induction Base:
+'(gen_0':s:cons_+2_3(0), gen_0':s:cons_+2_3(b)) ->_R^Omega(1)
gen_0':s:cons_+2_3(b)

Induction Step:
+'(gen_0':s:cons_+2_3(+(n4_3, 1)), gen_0':s:cons_+2_3(b)) ->_R^Omega(1)
s(+'(gen_0':s:cons_+2_3(n4_3), gen_0':s:cons_+2_3(b))) ->_IH
s(gen_0':s:cons_+2_3(+(b, c5_3)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(62)
Complex Obligation (BEST)

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(63)
Obligation:
Proved the lower bound n^1 for the following obligation:

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

Types:
+' :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
0' :: 0':s:cons_+
s :: 0':s:cons_+ -> 0':s:cons_+
encArg :: 0':s:cons_+ -> 0':s:cons_+
cons_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_0 :: 0':s:cons_+
encode_s :: 0':s:cons_+ -> 0':s:cons_+
hole_0':s:cons_+1_3 :: 0':s:cons_+
gen_0':s:cons_+2_3 :: Nat -> 0':s:cons_+


Generator Equations:
gen_0':s:cons_+2_3(0) <=> 0'
gen_0':s:cons_+2_3(+(x, 1)) <=> s(gen_0':s:cons_+2_3(x))


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

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

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

(65)
BOUNDS(n^1, INF)

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

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

Types:
+' :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
0' :: 0':s:cons_+
s :: 0':s:cons_+ -> 0':s:cons_+
encArg :: 0':s:cons_+ -> 0':s:cons_+
cons_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_+ :: 0':s:cons_+ -> 0':s:cons_+ -> 0':s:cons_+
encode_0 :: 0':s:cons_+
encode_s :: 0':s:cons_+ -> 0':s:cons_+
hole_0':s:cons_+1_3 :: 0':s:cons_+
gen_0':s:cons_+2_3 :: Nat -> 0':s:cons_+


Lemmas:
+'(gen_0':s:cons_+2_3(n4_3), gen_0':s:cons_+2_3(b)) -> gen_0':s:cons_+2_3(+(n4_3, b)), rt in Omega(1 + n4_3)


Generator Equations:
gen_0':s:cons_+2_3(0) <=> 0'
gen_0':s:cons_+2_3(+(x, 1)) <=> s(gen_0':s:cons_+2_3(x))


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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_+2_3(n751_3)) -> gen_0':s:cons_+2_3(n751_3), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_+2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_+2_3(+(n751_3, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_+2_3(n751_3))) ->_IH
s(gen_0':s:cons_+2_3(c752_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(68)
BOUNDS(1, INF)