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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 183 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
    (16) CpxRNTS
    (17) InliningProof [UPPER BOUND(ID), 497 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), 173 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 35 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 50 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 21 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 21 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 423 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 217 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 461 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 371 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 297 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 167 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (76) CpxRNTS
    (77) FinalProof [FINISHED, 0 ms]
    (78) BOUNDS(1, n^2)
(79) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(80) TRS for Loop Detection
    (81) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (82) BEST
        (83) proven lower bound
            (84) LowerBoundPropagationProof [FINISHED, 0 ms]
            (85) BOUNDS(n^1, INF)
        (86) TRS for Loop Detection


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

(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:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(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:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(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:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(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:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(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:

   not(true) -> false [1]
   not(false) -> true [1]
   evenodd(x, 0) -> not(evenodd(x, s(0))) [1]
   evenodd(0, s(0)) -> false [1]
   evenodd(s(x), s(0)) -> evenodd(x, 0) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_not(x_1)) -> not(encArg(x_1)) [0]
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_true -> true [0]
   encode_false -> false [0]
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

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

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

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

   not(true) -> false [1]
   not(false) -> true [1]
   evenodd(x, 0) -> not(evenodd(x, s(0))) [1]
   evenodd(0, s(0)) -> false [1]
   evenodd(s(x), s(0)) -> evenodd(x, 0) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_not(x_1)) -> not(encArg(x_1)) [0]
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_true -> true [0]
   encode_false -> false [0]
   encode_evenodd(x_1, x_2) -> evenodd(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:
not :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
true :: true:false:0:s:cons_not:cons_evenodd
false :: true:false:0:s:cons_not:cons_evenodd
evenodd :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
0 :: true:false:0:s:cons_not:cons_evenodd
s :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
encArg :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
cons_not :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
cons_evenodd :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
encode_not :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
encode_true :: true:false:0:s:cons_not:cons_evenodd
encode_false :: true:false:0:s:cons_not:cons_evenodd
encode_evenodd :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd
encode_0 :: true:false:0:s:cons_not:cons_evenodd
encode_s :: true:false:0:s:cons_not:cons_evenodd -> true:false:0:s:cons_not:cons_evenodd

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

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

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

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

(c) The following functions are completely defined:

   evenodd_2
   not_1
   encArg_1
   encode_not_1
   encode_true
   encode_false
   encode_evenodd_2
   encode_0
   encode_s_1

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   encode_evenodd(v0, v1) -> null_encode_evenodd [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   evenodd(v0, v1) -> null_evenodd [0]
   not(v0) -> null_not [0]

And the following fresh constants:    null_encArg, null_encode_not, null_encode_true, null_encode_false, null_encode_evenodd, null_encode_0, null_encode_s, null_evenodd, null_not

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

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

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

   not(true) -> false [1]
   not(false) -> true [1]
   evenodd(x, 0) -> not(evenodd(x, s(0))) [1]
   evenodd(0, s(0)) -> false [1]
   evenodd(s(x), s(0)) -> evenodd(x, 0) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_not(x_1)) -> not(encArg(x_1)) [0]
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_true -> true [0]
   encode_false -> false [0]
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   encode_evenodd(v0, v1) -> null_encode_evenodd [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   evenodd(v0, v1) -> null_evenodd [0]
   not(v0) -> null_not [0]

The TRS has the following type information:
not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
true :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
false :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
0 :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
s :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encArg :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
cons_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
cons_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_true :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_false :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_0 :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_s :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encArg :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_true :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_false :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_0 :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_s :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not

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

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

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

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

   not(true) -> false [1]
   not(false) -> true [1]
   evenodd(0, 0) -> not(false) [2]
   evenodd(s(x'), 0) -> not(evenodd(x', 0)) [2]
   evenodd(x, 0) -> not(null_evenodd) [1]
   evenodd(0, s(0)) -> false [1]
   evenodd(s(x), s(0)) -> evenodd(x, 0) [1]
   encArg(true) -> true [0]
   encArg(false) -> false [0]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_not(true)) -> not(true) [0]
   encArg(cons_not(false)) -> not(false) [0]
   encArg(cons_not(0)) -> not(0) [0]
   encArg(cons_not(s(x_1'))) -> not(s(encArg(x_1'))) [0]
   encArg(cons_not(cons_not(x_1''))) -> not(not(encArg(x_1''))) [0]
   encArg(cons_not(cons_evenodd(x_11, x_2'))) -> not(evenodd(encArg(x_11), encArg(x_2'))) [0]
   encArg(cons_not(x_1)) -> not(null_encArg) [0]
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2)) [0]
   encode_not(true) -> not(true) [0]
   encode_not(false) -> not(false) [0]
   encode_not(0) -> not(0) [0]
   encode_not(s(x_126)) -> not(s(encArg(x_126))) [0]
   encode_not(cons_not(x_127)) -> not(not(encArg(x_127))) [0]
   encode_not(cons_evenodd(x_128, x_28)) -> not(evenodd(encArg(x_128), encArg(x_28))) [0]
   encode_not(x_1) -> not(null_encArg) [0]
   encode_true -> true [0]
   encode_false -> false [0]
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_not(v0) -> null_encode_not [0]
   encode_true -> null_encode_true [0]
   encode_false -> null_encode_false [0]
   encode_evenodd(v0, v1) -> null_encode_evenodd [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   evenodd(v0, v1) -> null_evenodd [0]
   not(v0) -> null_not [0]

The TRS has the following type information:
not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
true :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
false :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
0 :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
s :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encArg :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
cons_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
cons_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_true :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_false :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_0 :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
encode_s :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not -> true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encArg :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_true :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_false :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_0 :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_encode_s :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_evenodd :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not
null_not :: true:false:0:s:cons_not:cons_evenodd:null_encArg:null_encode_not:null_encode_true:null_encode_false:null_encode_evenodd:null_encode_0:null_encode_s:null_evenodd:null_not

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

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

   true => 2
   false => 1
   0 => 0
   null_encArg => 0
   null_encode_not => 0
   null_encode_true => 0
   null_encode_false => 0
   null_encode_evenodd => 0
   null_encode_0 => 0
   null_encode_s => 0
   null_evenodd => 0
   null_not => 0

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

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

   encArg(z) -{ 0 }-> not(not(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(2) :|: z = 1 + 2
   encArg(z) -{ 0 }-> not(1) :|: z = 1 + 1
   encArg(z) -{ 0 }-> not(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> not(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> not(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   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_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_evenodd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(x_127))) :|: x_127 >= 0, z = 1 + x_127
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(2) :|: z = 2
   encode_not(z) -{ 0 }-> not(1) :|: z = 1
   encode_not(z) -{ 0 }-> not(0) :|: z = 0
   encode_not(z) -{ 0 }-> not(0) :|: x_1 >= 0, z = x_1
   encode_not(z) -{ 0 }-> not(1 + encArg(x_126)) :|: z = 1 + x_126, x_126 >= 0
   encode_not(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(x', 0)) :|: z = 1 + x', x' >= 0, z' = 0
   evenodd(z, z') -{ 2 }-> not(1) :|: z = 0, z' = 0
   evenodd(z, z') -{ 1 }-> not(0) :|: x >= 0, z = x, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(x, 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

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

   encArg(z) -{ 0 }-> not(not(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + x_1, x_1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_evenodd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(x_127))) :|: x_127 >= 0, z = 1 + x_127
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(x_126)) :|: z = 1 + x_126, x_126 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: x_1 >= 0, z = x_1, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(x', 0)) :|: z = 1 + x', x' >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(x, 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(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 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { not }
   { encode_0 }
   { encode_false }
   { encode_true }
   { evenodd }
   { encArg }
   { encode_not }
   { encode_evenodd }
   { encode_s }

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {not}, {encode_0}, {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {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 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {not}, {encode_0}, {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: not
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {not}, {encode_0}, {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: ?, size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: not
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_0}, {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]

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

(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 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_0}, {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]

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

(31) 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

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_0}, {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: ?, size: O(1) [0]

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

(33) 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

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_false}, {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: ?, size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]

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

(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) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_true
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_true}, {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: ?, size: O(1) [2]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]

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

(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) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: evenodd
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {evenodd}, {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: ?, size: O(1) [2]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: evenodd
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 3*z

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 2 }-> not(evenodd(z - 1, 0)) :|: z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 1 }-> evenodd(z - 1, 0) :|: z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]

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

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]

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

(55) 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: 2 + z

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: ?, size: O(n^1) [2 + z]

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

(57) 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 + 10*z + 3*z^2

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

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

   encArg(z) -{ 0 }-> not(not(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> not(evenodd(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 0 }-> not(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> evenodd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 0 }-> evenodd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ 0 }-> not(not(encArg(z - 1))) :|: z - 1 >= 0
   encode_not(z) -{ 0 }-> not(evenodd(encArg(x_128), encArg(x_28))) :|: x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 0 }-> not(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]

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

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_not
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_not}, {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: ?, size: O(1) [2]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_not
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 18 + 31*z + 12*z^2

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: O(n^2) [18 + 31*z + 12*z^2], size: O(1) [2]

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

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: O(n^2) [18 + 31*z + 12*z^2], size: O(1) [2]

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_evenodd
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_evenodd}, {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: O(n^2) [18 + 31*z + 12*z^2], size: O(1) [2]
  encode_evenodd: runtime: ?, size: O(1) [2]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_evenodd
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 15 + 13*z + 3*z^2 + 10*z' + 3*z'^2

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: O(n^2) [18 + 31*z + 12*z^2], size: O(1) [2]
  encode_evenodd: runtime: O(n^2) [15 + 13*z + 3*z^2 + 10*z' + 3*z'^2], size: O(1) [2]

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

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: O(n^2) [18 + 31*z + 12*z^2], size: O(1) [2]
  encode_evenodd: runtime: O(n^2) [15 + 13*z + 3*z^2 + 10*z' + 3*z'^2], size: O(1) [2]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: O(n^2) [18 + 31*z + 12*z^2], size: O(1) [2]
  encode_evenodd: runtime: O(n^2) [15 + 13*z + 3*z^2 + 10*z' + 3*z'^2], size: O(1) [2]
  encode_s: runtime: ?, size: O(n^1) [3 + z]

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

(75) 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 + 10*z + 3*z^2

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ -6 + -2*z + 3*z^2 }-> s10 :|: s9 >= 0, s9 <= z - 2 + 2, s10 >= 0, s10 <= 2, z - 2 >= 0
   encArg(z) -{ -5 + -2*z + 3*z^2 }-> s13 :|: s11 >= 0, s11 <= z - 2 + 2, s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, z - 2 >= 0
   encArg(z) -{ 10 + 3*s14 + 10*x_11 + 3*x_11^2 + 10*x_2' + 3*x_2'^2 }-> s17 :|: s14 >= 0, s14 <= x_11 + 2, s15 >= 0, s15 <= x_2' + 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2')
   encArg(z) -{ 9 + 3*s2 + 10*x_1 + 3*x_1^2 + 10*x_2 + 3*x_2^2 }-> s4 :|: s2 >= 0, s2 <= x_1 + 2, s3 >= 0, s3 <= x_2 + 2, s4 >= 0, s4 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 1 }-> 2 :|: z = 1 + 1, 1 = 1
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 1 + 2, 2 = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -6 + 4*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + 2, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_evenodd(z, z') -{ 9 + 3*s5 + 10*z + 3*z^2 + 10*z' + 3*z'^2 }-> s7 :|: s5 >= 0, s5 <= z + 2, s6 >= 0, s6 <= z' + 2, s7 >= 0, s7 <= 2, z >= 0, z' >= 0
   encode_evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_false -{ 0 }-> 1 :|: 
   encode_false -{ 0 }-> 0 :|: 
   encode_not(z) -{ -5 + 4*z + 3*z^2 }-> s19 :|: s18 >= 0, s18 <= z - 1 + 2, s19 >= 0, s19 <= 2, z - 1 >= 0
   encode_not(z) -{ -4 + 4*z + 3*z^2 }-> s22 :|: s20 >= 0, s20 <= z - 1 + 2, s21 >= 0, s21 <= 2, s22 >= 0, s22 <= 2, z - 1 >= 0
   encode_not(z) -{ 10 + 3*s23 + 10*x_128 + 3*x_128^2 + 10*x_28 + 3*x_28^2 }-> s26 :|: s23 >= 0, s23 <= x_128 + 2, s24 >= 0, s24 <= x_28 + 2, s25 >= 0, s25 <= 2, s26 >= 0, s26 <= 2, x_128 >= 0, z = 1 + x_128 + x_28, x_28 >= 0
   encode_not(z) -{ 1 }-> 2 :|: z = 1, 1 = 1
   encode_not(z) -{ 1 }-> 1 :|: z = 2, 2 = 2
   encode_not(z) -{ 0 }-> 0 :|: z >= 0
   encode_not(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0
   encode_not(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_not(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 1 + 10*z + 3*z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z + 2, z >= 0
   encode_true -{ 0 }-> 2 :|: 
   encode_true -{ 0 }-> 0 :|: 
   evenodd(z, z') -{ 5 + 3*z }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' = 1 + 0
   evenodd(z, z') -{ 7 + 3*z }-> s'' :|: s' >= 0, s' <= 2, s'' >= 0, s'' <= 2, z - 1 >= 0, z' = 0
   evenodd(z, z') -{ 3 }-> 2 :|: z = 0, z' = 0, 1 = 1
   evenodd(z, z') -{ 1 }-> 1 :|: z' = 1 + 0, z = 0
   evenodd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   evenodd(z, z') -{ 2 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
   evenodd(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0, v0 >= 0, 0 = v0
   not(z) -{ 1 }-> 2 :|: z = 1
   not(z) -{ 1 }-> 1 :|: z = 2
   not(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  not: runtime: O(1) [1], size: O(1) [2]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  encode_false: runtime: O(1) [0], size: O(1) [1]
  encode_true: runtime: O(1) [0], size: O(1) [2]
  evenodd: runtime: O(n^1) [7 + 3*z], size: O(1) [2]
  encArg: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [2 + z]
  encode_not: runtime: O(n^2) [18 + 31*z + 12*z^2], size: O(1) [2]
  encode_evenodd: runtime: O(n^2) [15 + 13*z + 3*z^2 + 10*z' + 3*z'^2], size: O(1) [2]
  encode_s: runtime: O(n^2) [1 + 10*z + 3*z^2], size: O(n^1) [3 + z]

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

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

(78)
BOUNDS(1, n^2)

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

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

(80)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

The rewrite sequence

evenodd(s(x1_0), 0) ->^+ not(evenodd(x1_0, 0))

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

The pumping substitution is [x1_0 / s(x1_0)].

The result substitution is [ ].




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

(82)
Complex Obligation (BEST)

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

(83)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

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

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

(85)
BOUNDS(n^1, INF)

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

(86)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_evenodd(x_1, x_2)) -> evenodd(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_evenodd(x_1, x_2) -> evenodd(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: FULL