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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 181 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) InliningProof [UPPER BOUND(ID), 576 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), 379 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1154 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 1127 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 397 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 277 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 385 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 68 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (76) CpxRNTS
    (77) FinalProof [FINISHED, 0 ms]
    (78) BOUNDS(1, n^3)
(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 (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

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

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


The TRS R consists of the following rules:

   -(x, 0) -> x [1]
   -(s(x), s(y)) -> -(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x)))) [1]
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]

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

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

- => minus

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

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


The TRS R consists of the following rules:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), y) -> f(p(minus(s(x), y)), p(minus(y, s(x)))) [1]
   f(x, s(y)) -> f(p(minus(x, s(y))), p(minus(s(y), x))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_-(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]

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

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

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

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), y) -> f(p(minus(s(x), y)), p(minus(y, s(x)))) [1]
   f(x, s(y)) -> f(p(minus(x, s(y))), p(minus(s(y), x))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_-(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
minus :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
0 :: 0:s:cons_-:cons_p:cons_f
s :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encArg :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_- :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_- :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_0 :: 0:s:cons_-:cons_p:cons_f
encode_s :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f

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:

   p_1
   minus_2
   f_2
   encArg_1
   encode_-_2
   encode_0
   encode_s_1
   encode_p_1
   encode_f_2

Due to the following rules being added:

   encArg(v0) -> 0 [0]
   encode_-(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   encode_p(v0) -> 0 [0]
   encode_f(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]
   minus(v0, v1) -> 0 [0]
   f(v0, v1) -> 0 [0]

And the following fresh constants: none

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

(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), y) -> f(p(minus(s(x), y)), p(minus(y, s(x)))) [1]
   f(x, s(y)) -> f(p(minus(x, s(y))), p(minus(s(y), x))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_-(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> 0 [0]
   encode_-(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   encode_p(v0) -> 0 [0]
   encode_f(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]
   minus(v0, v1) -> 0 [0]
   f(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
0 :: 0:s:cons_-:cons_p:cons_f
s :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encArg :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_- :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_- :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_0 :: 0:s:cons_-:cons_p:cons_f
encode_s :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f

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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), 0) -> f(p(s(x)), p(0)) [2]
   f(s(x), s(y')) -> f(p(minus(x, y')), p(minus(y', x))) [3]
   f(s(x), s(y')) -> f(p(minus(x, y')), p(0)) [2]
   f(s(x), s(x')) -> f(p(0), p(minus(x', x))) [2]
   f(s(x), y) -> f(p(0), p(0)) [1]
   f(s(x''), s(y)) -> f(p(minus(x'', y)), p(minus(y, x''))) [3]
   f(s(x''), s(y)) -> f(p(minus(x'', y)), p(0)) [2]
   f(0, s(y)) -> f(p(0), p(s(y))) [2]
   f(s(y''), s(y)) -> f(p(0), p(minus(y, y''))) [2]
   f(x, s(y)) -> f(p(0), p(0)) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_-(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_p(0)) -> p(0) [0]
   encArg(cons_p(s(x_127))) -> p(s(encArg(x_127))) [0]
   encArg(cons_p(cons_-(x_128, x_213))) -> p(minus(encArg(x_128), encArg(x_213))) [0]
   encArg(cons_p(cons_p(x_129))) -> p(p(encArg(x_129))) [0]
   encArg(cons_p(cons_f(x_130, x_214))) -> p(f(encArg(x_130), encArg(x_214))) [0]
   encArg(cons_p(x_1)) -> p(0) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_-(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_p(0) -> p(0) [0]
   encode_p(s(x_187)) -> p(s(encArg(x_187))) [0]
   encode_p(cons_-(x_188, x_243)) -> p(minus(encArg(x_188), encArg(x_243))) [0]
   encode_p(cons_p(x_189)) -> p(p(encArg(x_189))) [0]
   encode_p(cons_f(x_190, x_244)) -> p(f(encArg(x_190), encArg(x_244))) [0]
   encode_p(x_1) -> p(0) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> 0 [0]
   encode_-(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   encode_p(v0) -> 0 [0]
   encode_f(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]
   minus(v0, v1) -> 0 [0]
   f(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
0 :: 0:s:cons_-:cons_p:cons_f
s :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encArg :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_- :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
cons_f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_- :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_0 :: 0:s:cons_-:cons_p:cons_f
encode_s :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_p :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f
encode_f :: 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f -> 0:s:cons_-:cons_p:cons_f

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:

   0 => 0

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

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

   encArg(z) -{ 0 }-> p(p(encArg(x_129))) :|: z = 1 + (1 + x_129), x_129 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> p(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(x_127)) :|: z = 1 + (1 + x_127), x_127 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_-(z, z') -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_-(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_p(z) -{ 0 }-> p(p(encArg(x_189))) :|: z = 1 + x_189, x_189 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(0) :|: z = 0
   encode_p(z) -{ 0 }-> p(0) :|: x_1 >= 0, z = x_1
   encode_p(z) -{ 0 }-> p(1 + encArg(x_187)) :|: x_187 >= 0, z = 1 + x_187
   encode_p(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
   f(z, z') -{ 3 }-> f(p(minus(x, y')), p(minus(y', x))) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   f(z, z') -{ 2 }-> f(p(minus(x, y')), p(0)) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   f(z, z') -{ 3 }-> f(p(minus(x'', y)), p(minus(y, x''))) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0
   f(z, z') -{ 2 }-> f(p(minus(x'', y)), p(0)) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0
   f(z, z') -{ 2 }-> f(p(0), p(minus(x', x))) :|: z' = 1 + x', x >= 0, x' >= 0, z = 1 + x
   f(z, z') -{ 2 }-> f(p(0), p(minus(y, y''))) :|: z' = 1 + y, z = 1 + y'', y >= 0, y'' >= 0
   f(z, z') -{ 1 }-> f(p(0), p(0)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   f(z, z') -{ 1 }-> f(p(0), p(0)) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   f(z, z') -{ 2 }-> f(p(0), p(1 + y)) :|: z' = 1 + y, y >= 0, z = 0
   f(z, z') -{ 2 }-> f(p(1 + x), p(0)) :|: x >= 0, z = 1 + x, z' = 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

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

   encArg(z) -{ 0 }-> p(p(encArg(x_129))) :|: z = 1 + (1 + x_129), x_129 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(x_127)) :|: z = 1 + (1 + x_127), x_127 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   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_-(z, z') -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_-(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_p(z) -{ 0 }-> p(p(encArg(x_189))) :|: z = 1 + x_189, x_189 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(x_187)) :|: x_187 >= 0, z = 1 + x_187
   encode_p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: x >= 0, z = 1 + x, z' = 0, x' >= 0, 1 + x = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(x, y')), p(minus(y', x))) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   f(z, z') -{ 2 }-> f(p(minus(x, y')), 0) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(x'', y)), p(minus(y, x''))) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0
   f(z, z') -{ 2 }-> f(p(minus(x'', y)), 0) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' = 1 + y, y >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + y = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(x', x))) :|: z' = 1 + x', x >= 0, x' >= 0, z = 1 + x, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, p(minus(y, y''))) :|: z' = 1 + y, z = 1 + y'', y >= 0, y'' >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: x >= 0, z = 1 + x, z' = 0, v0 >= 0, 1 + x = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' = 1 + y, y >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + y = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(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 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0


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

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

   { minus }
   { encode_0 }
   { p }
   { f }
   { encArg }
   { encode_p }
   { encode_f }
   { encode_- }
   { encode_s }

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {encode_0}, {p}, {f}, {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {encode_0}, {p}, {f}, {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {encode_0}, {p}, {f}, {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_0}, {p}, {f}, {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_0}, {p}, {f}, {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

(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 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_0}, {p}, {f}, {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {f}, {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  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 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 5 + z + z' }-> f(s3, s4) :|: s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 3 + z' }-> f(s5, 0) :|: s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z }-> f(0, s6) :|: s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 5 + z + z' }-> f(s3, s4) :|: s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 3 + z' }-> f(s5, 0) :|: s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z }-> f(0, s6) :|: s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 17 + 8*z + z*z' + z^2 + 8*z' + z'^2

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 5 + z + z' }-> f(s3, s4) :|: s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 3 + z' }-> f(s5, 0) :|: s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z }-> f(0, s6) :|: s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]

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

(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 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: ?, size: O(n^1) [z]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 41 + 116*z + 74*z^2 + 12*z^3

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

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

   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(minus(encArg(x_128), encArg(x_213))) :|: x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 0 }-> p(f(encArg(x_130), encArg(x_214))) :|: x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   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_-(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(minus(encArg(x_188), encArg(x_243))) :|: z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ 0 }-> p(f(encArg(x_190), encArg(x_244))) :|: x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(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
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]

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

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_p}, {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: ?, size: O(n^1) [z]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_p
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 184 + 489*z + 375*z^2 + 72*z^3

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [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) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: ?, size: O(1) [0]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: O(n^3) [99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3], size: O(1) [0]

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

(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) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: O(n^3) [99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3], size: O(1) [0]

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_-}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: O(n^3) [99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3], size: O(1) [0]
  encode_-: runtime: ?, size: O(n^1) [z]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_-
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 83 + 116*z + 74*z^2 + 12*z^3 + 117*z' + 74*z'^2 + 12*z'^3

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: O(n^3) [99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3], size: O(1) [0]
  encode_-: runtime: O(n^3) [83 + 116*z + 74*z^2 + 12*z^3 + 117*z' + 74*z'^2 + 12*z'^3], size: O(n^1) [z]

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

(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) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: O(n^3) [99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3], size: O(1) [0]
  encode_-: runtime: O(n^3) [83 + 116*z + 74*z^2 + 12*z^3 + 117*z' + 74*z'^2 + 12*z'^3], size: O(n^1) [z]

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

(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: 1 + z

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: O(n^3) [99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3], size: O(1) [0]
  encode_-: runtime: O(n^3) [83 + 116*z + 74*z^2 + 12*z^3 + 117*z' + 74*z'^2 + 12*z'^3], size: O(n^1) [z]
  encode_s: runtime: ?, size: O(n^1) [1 + 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^3) with polynomial bound: 41 + 116*z + 74*z^2 + 12*z^3

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

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

   encArg(z) -{ 83 + s18 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s19 :|: s17 >= 0, s17 <= x_1, s18 >= 0, s18 <= x_2, s19 >= 0, s19 <= s17, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 99 + 8*s20 + s20*s21 + s20^2 + 8*s21 + s21^2 + 116*x_1 + 74*x_1^2 + 12*x_1^3 + 116*x_2 + 74*x_2^2 + 12*x_2^3 }-> s22 :|: s20 >= 0, s20 <= x_1, s21 >= 0, s21 <= x_2, s22 >= 0, s22 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 10 + -36*z + 2*z^2 + 12*z^3 }-> s31 :|: s30 >= 0, s30 <= z - 2, s31 >= 0, s31 <= 1 + s30, z - 2 >= 0
   encArg(z) -{ 84 + s33 + 116*x_128 + 74*x_128^2 + 12*x_128^3 + 116*x_213 + 74*x_213^2 + 12*x_213^3 }-> s35 :|: s32 >= 0, s32 <= x_128, s33 >= 0, s33 <= x_213, s34 >= 0, s34 <= s32, s35 >= 0, s35 <= s34, x_128 >= 0, x_213 >= 0, z = 1 + (1 + x_128 + x_213)
   encArg(z) -{ 11 + -36*z + 2*z^2 + 12*z^3 }-> s38 :|: s36 >= 0, s36 <= z - 2, s37 >= 0, s37 <= s36, s38 >= 0, s38 <= s37, z - 2 >= 0
   encArg(z) -{ 100 + 8*s39 + s39*s40 + s39^2 + 8*s40 + s40^2 + 116*x_130 + 74*x_130^2 + 12*x_130^3 + 116*x_214 + 74*x_214^2 + 12*x_214^3 }-> s42 :|: s39 >= 0, s39 <= x_130, s40 >= 0, s40 <= x_214, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= s41, x_130 >= 0, z = 1 + (1 + x_130 + x_214), x_214 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0
   encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0
   encArg(z) -{ -13 + 4*z + 38*z^2 + 12*z^3 }-> 1 + s16 :|: s16 >= 0, s16 <= z - 1, z - 1 >= 0
   encode_-(z, z') -{ 83 + s24 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_-(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 99 + 8*s27 + s27*s28 + s27^2 + 8*s28 + s28^2 + 116*z + 74*z^2 + 12*z^3 + 116*z' + 74*z'^2 + 12*z'^3 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_p(z) -{ -12 + 4*z + 38*z^2 + 12*z^3 }-> s44 :|: s43 >= 0, s43 <= z - 1, s44 >= 0, s44 <= 1 + s43, z - 1 >= 0
   encode_p(z) -{ 84 + s46 + 116*x_188 + 74*x_188^2 + 12*x_188^3 + 116*x_243 + 74*x_243^2 + 12*x_243^3 }-> s48 :|: s45 >= 0, s45 <= x_188, s46 >= 0, s46 <= x_243, s47 >= 0, s47 <= s45, s48 >= 0, s48 <= s47, z = 1 + x_188 + x_243, x_243 >= 0, x_188 >= 0
   encode_p(z) -{ -11 + 4*z + 38*z^2 + 12*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0
   encode_p(z) -{ 100 + 8*s52 + s52*s53 + s52^2 + 8*s53 + s53^2 + 116*x_190 + 74*x_190^2 + 12*x_190^3 + 116*x_244 + 74*x_244^2 + 12*x_244^3 }-> s55 :|: s52 >= 0, s52 <= x_190, s53 >= 0, s53 <= x_244, s54 >= 0, s54 <= 0, s55 >= 0, s55 <= s54, x_244 >= 0, x_190 >= 0, z = 1 + x_190 + x_244
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 41 + 116*z + 74*z^2 + 12*z^3 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z') -{ 20 + 8*s5 + s5^2 + z' }-> s10 :|: s10 >= 0, s10 <= 0, s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 20 + 8*s6 + s6^2 + z }-> s11 :|: s11 >= 0, s11 <= 0, s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 18 }-> s12 :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x + x^2 }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 19 }-> s14 :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 18 }-> s15 :|: s15 >= 0, s15 <= 0, z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 20 + 8*x' + x'^2 }-> s7 :|: s7 >= 0, s7 <= 0, z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 19 }-> s8 :|: s8 >= 0, s8 <= 0, z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 22 + 8*s3 + s3*s4 + s3^2 + 8*s4 + s4^2 + z + z' }-> s9 :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]
  encArg: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [z]
  encode_p: runtime: O(n^3) [184 + 489*z + 375*z^2 + 72*z^3], size: O(n^1) [z]
  encode_f: runtime: O(n^3) [99 + 124*z + z*z' + 75*z^2 + 12*z^3 + 124*z' + 75*z'^2 + 12*z'^3], size: O(1) [0]
  encode_-: runtime: O(n^3) [83 + 116*z + 74*z^2 + 12*z^3 + 117*z' + 74*z'^2 + 12*z'^3], size: O(n^1) [z]
  encode_s: runtime: O(n^3) [41 + 116*z + 74*z^2 + 12*z^3], size: O(n^1) [1 + z]

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

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

(78)
BOUNDS(1, n^3)

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

(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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

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

The rewrite sequence

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

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

The pumping substitution is [x / s(x), y / s(y)].

The result substitution is [ ].




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

(82)
Complex Obligation (BEST)

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

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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

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

(85)
BOUNDS(n^1, INF)

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

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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST