/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^4))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^4).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 280 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 390 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 5 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 451 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 78 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 11 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 817 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 438 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 482 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 201 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 209 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 239 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (72) CpxRNTS
    (73) FinalProof [FINISHED, 0 ms]
    (74) BOUNDS(1, n^4)
(75) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(76) TRS for Loop Detection
    (77) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (78) BEST
        (79) proven lower bound
            (80) LowerBoundPropagationProof [FINISHED, 0 ms]
            (81) BOUNDS(n^1, INF)
        (82) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^4).


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

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_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))


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

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


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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^4).


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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^4).


The TRS R consists of the following rules:

   minus(x, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_minus(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_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(x, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_minus(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_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]

The TRS has the following type information:
minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
0 :: 0:s:cons_minus:cons_div:cons_f
s :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encArg :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_0 :: 0:s:cons_minus:cons_div:cons_f
encode_s :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f

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

(9) 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:

   minus_2
   f_3
   div_2
   encArg_1
   encode_minus_2
   encode_0
   encode_s_1
   encode_div_2
   encode_f_3

Due to the following rules being added:

   encArg(v0) -> 0 [0]
   encode_minus(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   encode_div(v0, v1) -> 0 [0]
   encode_f(v0, v1, v2) -> 0 [0]
   minus(v0, v1) -> 0 [0]
   f(v0, v1, v2) -> 0 [0]
   div(v0, v1) -> 0 [0]

And the following fresh constants: none

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

(10)
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, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_minus(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_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(v0) -> 0 [0]
   encode_minus(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   encode_div(v0, v1) -> 0 [0]
   encode_f(v0, v1, v2) -> 0 [0]
   minus(v0, v1) -> 0 [0]
   f(v0, v1, v2) -> 0 [0]
   div(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
0 :: 0:s:cons_minus:cons_div:cons_f
s :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encArg :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_0 :: 0:s:cons_minus:cons_div:cons_f
encode_s :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f

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

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

(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, x) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   minus(0, x) -> 0 [1]
   minus(x, 0) -> x [1]
   div(s(x), s(x)) -> s(div(0, s(x))) [2]
   div(s(s(x')), s(s(y'))) -> s(div(minus(x', y'), s(s(y')))) [2]
   div(s(0), s(y)) -> s(div(0, s(y))) [2]
   div(s(x), s(0)) -> s(div(x, s(0))) [2]
   div(s(x), s(y)) -> s(div(0, s(y))) [1]
   div(0, s(y)) -> 0 [1]
   f(x, 0, b) -> x [1]
   f(x, s(0), b) -> div(f(x, 0, b), b) [2]
   f(x, s(y), b) -> div(f(x, minus(y, 0), b), b) [2]
   f(x, s(y), b) -> div(f(x, 0, b), b) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_minus(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_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(v0) -> 0 [0]
   encode_minus(v0, v1) -> 0 [0]
   encode_0 -> 0 [0]
   encode_s(v0) -> 0 [0]
   encode_div(v0, v1) -> 0 [0]
   encode_f(v0, v1, v2) -> 0 [0]
   minus(v0, v1) -> 0 [0]
   f(v0, v1, v2) -> 0 [0]
   div(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
0 :: 0:s:cons_minus:cons_div:cons_f
s :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encArg :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
cons_f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_minus :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_0 :: 0:s:cons_minus:cons_div:cons_f
encode_s :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_div :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f
encode_f :: 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f -> 0:s:cons_minus:cons_div:cons_f

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

(13) 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

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   div(z, z') -{ 2 }-> 1 + div(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
   div(z, z') -{ 2 }-> 1 + div(minus(x', y'), 1 + (1 + y')) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   div(z, z') -{ 2 }-> 1 + div(0, 1 + x) :|: z' = 1 + x, x >= 0, z = 1 + x
   div(z, z') -{ 2 }-> 1 + div(0, 1 + y) :|: z' = 1 + y, z = 1 + 0, y >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_f(z, z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   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', z'') -{ 1 }-> x :|: b >= 0, z'' = b, x >= 0, z = x, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(x, minus(y, 0), b), b) :|: z' = 1 + y, b >= 0, z'' = b, x >= 0, y >= 0, z = x
   f(z, z', z'') -{ 2 }-> div(f(x, 0, b), b) :|: b >= 0, z'' = b, x >= 0, z' = 1 + 0, z = x
   f(z, z', z'') -{ 1 }-> div(f(x, 0, b), b) :|: z' = 1 + y, b >= 0, z'' = b, x >= 0, y >= 0, z = x
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   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') -{ 1 }-> 0 :|: z' = x, x >= 0, z = x
   minus(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { minus }
   { encode_0 }
   { div }
   { f }
   { encArg }
   { encode_div }
   { encode_minus }
   { encode_f }
   { encode_s }

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {encode_0}, {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {encode_0}, {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}

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

(21) 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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 2 }-> 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 2 }-> div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]

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

(27) 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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_0}, {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: ?, size: O(1) [0]

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

(29) 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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}, {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: ?, size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 10*z + z^2

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 + z }-> 1 + div(s', 1 + (1 + (z' - 2))) :|: s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 2 }-> 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ 1 }-> 1 + div(0, 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   div(z, z') -{ 2 }-> 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f}, {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: ?, size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 4 + z' }-> div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 2 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 1 }-> div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: ?, size: O(n^1) [z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 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), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> div(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 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 0 }-> div(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 0 }-> minus(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_div}, {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: ?, size: O(n^1) [z]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_minus}, {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: ?, size: O(n^1) [z]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: O(n^4) [13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: O(n^4) [13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}, {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: O(n^4) [13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_f: runtime: ?, size: O(n^1) [z]

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 16 + 36*z + 30*z*z' + 95*z^2 + 3*z^2*z' + 99*z^3 + 9*z^4 + 46*z' + 96*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: O(n^4) [13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_f: runtime: O(n^4) [16 + 36*z + 30*z*z' + 95*z^2 + 3*z^2*z' + 99*z^3 + 9*z^4 + 46*z' + 96*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4], size: O(n^1) [z]

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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: O(n^4) [13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_f: runtime: O(n^4) [16 + 36*z + 30*z*z' + 95*z^2 + 3*z^2*z' + 99*z^3 + 9*z^4 + 46*z' + 96*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4], size: O(n^1) [z]

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

(69) 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

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_s}
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: O(n^4) [13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_f: runtime: O(n^4) [16 + 36*z + 30*z*z' + 95*z^2 + 3*z^2*z' + 99*z^3 + 9*z^4 + 46*z' + 96*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4], size: O(n^1) [z]
  encode_s: runtime: ?, size: O(n^1) [1 + z]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_s
after applying outer abstraction to obtain an ITS,
resulting in: O(n^4) with polynomial bound: 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4

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

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

   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   div(z, z') -{ 3 }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z = 1 + (z' - 1)
   div(z, z') -{ 4 + 10*s' + s'^2 + z }-> 1 + s2 :|: s2 >= 0, s2 <= s', s' >= 0, s' <= z - 2, z - 2 >= 0, z' - 2 >= 0
   div(z, z') -{ 3 }-> 1 + s3 :|: s3 >= 0, s3 <= 0, z = 1 + 0, z' - 1 >= 0
   div(z, z') -{ -6 + 8*z + z^2 }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1, z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 2 }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 1 >= 0, z' - 1 >= 0
   encArg(z) -{ 13 + s13 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s15 :|: s13 >= 0, s13 <= x_1, s14 >= 0, s14 <= x_2, s15 >= 0, s15 <= s13, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 11 + 10*s16 + s16^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 }-> s18 :|: s16 >= 0, s16 <= x_1, s17 >= 0, s17 <= x_2, s18 >= 0, s18 <= s16, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 16 + 30*s19*s20 + 3*s19^2*s20 + 10*s20 + s20^2 + 36*x_1 + 95*x_1^2 + 99*x_1^3 + 9*x_1^4 + 36*x_2 + 95*x_2^2 + 99*x_2^3 + 9*x_2^4 + 36*x_3 + 95*x_3^2 + 99*x_3^3 + 9*x_3^4 }-> s22 :|: s19 >= 0, s19 <= x_1, s20 >= 0, s20 <= x_2, s21 >= 0, s21 <= x_3, s22 >= 0, s22 <= s19, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -26 + 107*z + -148*z^2 + 63*z^3 + 9*z^4 }-> 1 + s12 :|: s12 >= 0, s12 <= z - 1, z - 1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_div(z, z') -{ 11 + 10*s27 + s27^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s29 :|: s27 >= 0, s27 <= z, s28 >= 0, s28 <= z', s29 >= 0, s29 <= s27, z >= 0, z' >= 0
   encode_div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_f(z, z', z'') -{ 16 + 30*s30*s31 + 3*s30^2*s31 + 10*s31 + s31^2 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4 }-> s33 :|: s30 >= 0, s30 <= z, s31 >= 0, s31 <= z', s32 >= 0, s32 <= z'', s33 >= 0, s33 <= s30, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_minus(z, z') -{ 13 + s23 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4 }-> s25 :|: s23 >= 0, s23 <= z, s24 >= 0, s24 <= z', s25 >= 0, s25 <= s23, z >= 0, z' >= 0
   encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_s(z) -{ 0 }-> 0 :|: z >= 0
   encode_s(z) -{ 5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4 }-> 1 + s26 :|: s26 >= 0, s26 <= z, z >= 0
   f(z, z', z'') -{ 3 + 10*s10 + s10^2 }-> s11 :|: s10 >= 0, s10 <= z, s11 >= 0, s11 <= s10, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 4 + 10*s6 + s6^2 }-> s7 :|: s6 >= 0, s6 <= z, s7 >= 0, s7 <= s6, z'' >= 0, z >= 0, z' = 1 + 0
   f(z, z', z'') -{ 6 + 10*s'' + 30*s''*z + 3*s''*z^2 + s''^2 + 10*s8 + s8^2 + z' }-> s9 :|: s8 >= 0, s8 <= z, s9 >= 0, s9 <= s8, s'' >= 0, s'' <= z' - 1, z'' >= 0, z >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 1 }-> z :|: z'' >= 0, z >= 0, z' = 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   minus(z, z') -{ 3 + 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') -{ 1 }-> 0 :|: z' >= 0, z = z'
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [3 + z], size: O(n^1) [z]
  encode_0: runtime: O(1) [0], size: O(1) [0]
  div: runtime: O(n^2) [1 + 10*z + z^2], size: O(n^1) [z]
  f: runtime: O(n^3) [1 + 30*z*z' + 3*z^2*z' + 10*z' + z'^2], size: O(n^1) [z]
  encArg: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [z]
  encode_div: runtime: O(n^4) [11 + 46*z + 96*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_minus: runtime: O(n^4) [13 + 37*z + 95*z^2 + 99*z^3 + 9*z^4 + 36*z' + 95*z'^2 + 99*z'^3 + 9*z'^4], size: O(n^1) [z]
  encode_f: runtime: O(n^4) [16 + 36*z + 30*z*z' + 95*z^2 + 3*z^2*z' + 99*z^3 + 9*z^4 + 46*z' + 96*z'^2 + 99*z'^3 + 9*z'^4 + 36*z'' + 95*z''^2 + 99*z''^3 + 9*z''^4], size: O(n^1) [z]
  encode_s: runtime: O(n^4) [5 + 36*z + 95*z^2 + 99*z^3 + 9*z^4], size: O(n^1) [1 + z]

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

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

(74)
BOUNDS(1, n^4)

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

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

(76)
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^4).


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

The rewrite sequence

minus(s(x), s(y)) ->^+ minus(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 [ ].




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

(78)
Complex Obligation (BEST)

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

(79)
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^4).


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(81)
BOUNDS(n^1, INF)

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

(82)
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^4).


The TRS R consists of the following rules:

   minus(x, x) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   minus(0, x) -> 0
   minus(x, 0) -> x
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(0, s(y)) -> 0
   f(x, 0, b) -> x
   f(x, s(y), b) -> div(f(x, minus(s(y), s(0)), b), b)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST