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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 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), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 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), 97 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 192 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 333 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 700 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 341 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 457 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 169 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), 209 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), 325 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) FinalProof [FINISHED, 0 ms]
    (68) BOUNDS(1, n^2)
(69) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(70) TRS for Loop Detection
    (71) DecreasingLoopProof [LOWER BOUND(ID), 8 ms]
    (72) BEST
        (73) proven lower bound
            (74) LowerBoundPropagationProof [FINISHED, 0 ms]
            (75) BOUNDS(n^1, INF)
        (76) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))


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

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


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

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

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

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


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

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^2).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a))) [1]
   mark(f(X)) -> a__f(X) [1]
   mark(a) -> a [1]
   mark(g(X)) -> g(mark(X)) [1]
   a__f(X) -> f(X) [1]
   encArg(f(x_1)) -> f(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) [0]
   encArg(cons_mark(x_1)) -> mark(encArg(x_1)) [0]
   encode_a__f(x_1) -> a__f(encArg(x_1)) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_mark(x_1) -> mark(encArg(x_1)) [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:

   a__f(f(a)) -> a__f(g(f(a))) [1]
   mark(f(X)) -> a__f(X) [1]
   mark(a) -> a [1]
   mark(g(X)) -> g(mark(X)) [1]
   a__f(X) -> f(X) [1]
   encArg(f(x_1)) -> f(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) [0]
   encArg(cons_mark(x_1)) -> mark(encArg(x_1)) [0]
   encode_a__f(x_1) -> a__f(encArg(x_1)) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_mark(x_1) -> mark(encArg(x_1)) [0]

The TRS has the following type information:
a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
a :: a:f:g:cons_a__f:cons_mark
g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encArg :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a :: a:f:g:cons_a__f:cons_mark
encode_g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark

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:

   a__f_1
   mark_1
   encArg_1
   encode_a__f_1
   encode_f_1
   encode_a
   encode_g_1
   encode_mark_1

Due to the following rules being added:

   encArg(v0) -> a [0]
   encode_a__f(v0) -> a [0]
   encode_f(v0) -> a [0]
   encode_a -> a [0]
   encode_g(v0) -> a [0]
   encode_mark(v0) -> a [0]
   mark(v0) -> a [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:

   a__f(f(a)) -> a__f(g(f(a))) [1]
   mark(f(X)) -> a__f(X) [1]
   mark(a) -> a [1]
   mark(g(X)) -> g(mark(X)) [1]
   a__f(X) -> f(X) [1]
   encArg(f(x_1)) -> f(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) [0]
   encArg(cons_mark(x_1)) -> mark(encArg(x_1)) [0]
   encode_a__f(x_1) -> a__f(encArg(x_1)) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_mark(x_1) -> mark(encArg(x_1)) [0]
   encArg(v0) -> a [0]
   encode_a__f(v0) -> a [0]
   encode_f(v0) -> a [0]
   encode_a -> a [0]
   encode_g(v0) -> a [0]
   encode_mark(v0) -> a [0]
   mark(v0) -> a [0]

The TRS has the following type information:
a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
a :: a:f:g:cons_a__f:cons_mark
g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encArg :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a :: a:f:g:cons_a__f:cons_mark
encode_g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark

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:

   a__f(f(a)) -> a__f(g(f(a))) [1]
   mark(f(X)) -> a__f(X) [1]
   mark(a) -> a [1]
   mark(g(X)) -> g(mark(X)) [1]
   a__f(X) -> f(X) [1]
   encArg(f(x_1)) -> f(encArg(x_1)) [0]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_a__f(f(x_1'))) -> a__f(f(encArg(x_1'))) [0]
   encArg(cons_a__f(a)) -> a__f(a) [0]
   encArg(cons_a__f(g(x_1''))) -> a__f(g(encArg(x_1''))) [0]
   encArg(cons_a__f(cons_a__f(x_11))) -> a__f(a__f(encArg(x_11))) [0]
   encArg(cons_a__f(cons_mark(x_12))) -> a__f(mark(encArg(x_12))) [0]
   encArg(cons_a__f(x_1)) -> a__f(a) [0]
   encArg(cons_mark(f(x_13))) -> mark(f(encArg(x_13))) [0]
   encArg(cons_mark(a)) -> mark(a) [0]
   encArg(cons_mark(g(x_14))) -> mark(g(encArg(x_14))) [0]
   encArg(cons_mark(cons_a__f(x_15))) -> mark(a__f(encArg(x_15))) [0]
   encArg(cons_mark(cons_mark(x_16))) -> mark(mark(encArg(x_16))) [0]
   encArg(cons_mark(x_1)) -> mark(a) [0]
   encode_a__f(f(x_17)) -> a__f(f(encArg(x_17))) [0]
   encode_a__f(a) -> a__f(a) [0]
   encode_a__f(g(x_18)) -> a__f(g(encArg(x_18))) [0]
   encode_a__f(cons_a__f(x_19)) -> a__f(a__f(encArg(x_19))) [0]
   encode_a__f(cons_mark(x_110)) -> a__f(mark(encArg(x_110))) [0]
   encode_a__f(x_1) -> a__f(a) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_mark(f(x_111)) -> mark(f(encArg(x_111))) [0]
   encode_mark(a) -> mark(a) [0]
   encode_mark(g(x_112)) -> mark(g(encArg(x_112))) [0]
   encode_mark(cons_a__f(x_113)) -> mark(a__f(encArg(x_113))) [0]
   encode_mark(cons_mark(x_114)) -> mark(mark(encArg(x_114))) [0]
   encode_mark(x_1) -> mark(a) [0]
   encArg(v0) -> a [0]
   encode_a__f(v0) -> a [0]
   encode_f(v0) -> a [0]
   encode_a -> a [0]
   encode_g(v0) -> a [0]
   encode_mark(v0) -> a [0]
   mark(v0) -> a [0]

The TRS has the following type information:
a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
a :: a:f:g:cons_a__f:cons_mark
g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encArg :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
cons_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a__f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_f :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_a :: a:f:g:cons_a__f:cons_mark
encode_g :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark
encode_mark :: a:f:g:cons_a__f:cons_mark -> a:f:g:cons_a__f:cons_mark

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:

   a => 0

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

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

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   encArg(z) -{ 0 }-> mark(mark(encArg(x_16))) :|: x_16 >= 0, z = 1 + (1 + x_16)
   encArg(z) -{ 0 }-> mark(a__f(encArg(x_15))) :|: x_15 >= 0, z = 1 + (1 + x_15)
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(x_13)) :|: z = 1 + (1 + x_13), x_13 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(x_14)) :|: x_14 >= 0, z = 1 + (1 + x_14)
   encArg(z) -{ 0 }-> a__f(mark(encArg(x_12))) :|: z = 1 + (1 + x_12), x_12 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11)
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(x_1'')) :|: z = 1 + (1 + x_1''), x_1'' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(x_110))) :|: z = 1 + x_110, x_110 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(x_19))) :|: z = 1 + x_19, x_19 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: x_1 >= 0, z = x_1
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(x_17)) :|: x_17 >= 0, z = 1 + x_17
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(x_18)) :|: z = 1 + x_18, x_18 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_f(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_g(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_mark(z) -{ 0 }-> mark(mark(encArg(x_114))) :|: x_114 >= 0, z = 1 + x_114
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(x_113))) :|: z = 1 + x_113, x_113 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: x_1 >= 0, z = x_1
   encode_mark(z) -{ 0 }-> mark(1 + encArg(x_111)) :|: z = 1 + x_111, x_111 >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(x_112)) :|: z = 1 + x_112, x_112 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   mark(z) -{ 1 }-> a__f(X) :|: z = 1 + X, X >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0


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

(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:

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0


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

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

   { encode_a }
   { a__f }
   { mark }
   { encArg }
   { encode_mark }
   { encode_g }
   { encode_f }
   { encode_a__f }

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

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

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {encode_a}, {a__f}, {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}

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

(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:

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {encode_a}, {a__f}, {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {encode_a}, {a__f}, {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {a__f}, {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]

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

(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:

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {a__f}, {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {a__f}, {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: ?, size: O(n^1) [3 + z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 1 }-> a__f(1 + (1 + 0)) :|: z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> a__f(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z = 0
   encode_a__f(z) -{ 0 }-> a__f(0) :|: z >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> a__f(z - 1) :|: z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]

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

(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:

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {mark}, {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: ?, size: O(n^1) [2 + z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + z

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(0) :|: z = 1 + 0
   encArg(z) -{ 0 }-> mark(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z = 0
   encode_mark(z) -{ 0 }-> mark(0) :|: z >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + 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:

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encArg}, {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: ?, size: O(n^1) [3*z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 8*z + 6*z^2

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> mark(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> mark(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(mark(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(a__f(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a__f(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 0 }-> a__f(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> a__f(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> mark(mark(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(a__f(encArg(z - 1))) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> mark(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*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:

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: encode_mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 3*z

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_mark}, {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: ?, size: O(n^1) [2 + 3*z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 45 + 2*z + 18*z^2

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*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:

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_g}, {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: ?, size: O(n^1) [1 + 3*z]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 8*z + 6*z^2

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*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:

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]

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

(57) 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: 1 + 3*z

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_f}, {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]
  encode_f: runtime: ?, size: O(n^1) [1 + 3*z]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 8*z + 6*z^2

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]
  encode_f: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*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:

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]
  encode_f: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: {encode_a__f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]
  encode_f: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]
  encode_a__f: runtime: ?, size: O(n^1) [3 + 3*z]

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_a__f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 34 + 18*z^2

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

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

   a__f(z) -{ 3 }-> s :|: s >= 0, s <= 1 + (1 + 0) + 3, z = 1 + 0
   a__f(z) -{ 1 }-> 1 + z :|: z >= 0
   encArg(z) -{ 2 }-> s'' :|: s'' >= 0, s'' <= 0 + 3, z = 1 + 0
   encArg(z) -{ 2 }-> s1 :|: s1 >= 0, s1 <= 0 + 3, z - 1 >= 0
   encArg(z) -{ 18 + -16*z + 6*z^2 }-> s11 :|: s10 >= 0, s10 <= 3 * (z - 2), s11 >= 0, s11 <= 1 + s10 + 3, z - 2 >= 0
   encArg(z) -{ 20 + -16*z + 6*z^2 }-> s14 :|: s12 >= 0, s12 <= 3 * (z - 2), s13 >= 0, s13 <= s12 + 3, s14 >= 0, s14 <= s13 + 3, z - 2 >= 0
   encArg(z) -{ 22 + s15 + -16*z + 6*z^2 }-> s17 :|: s15 >= 0, s15 <= 3 * (z - 2), s16 >= 0, s16 <= s15 + 2, s17 >= 0, s17 <= s16 + 3, z - 2 >= 0
   encArg(z) -{ 21 + s18 + -16*z + 6*z^2 }-> s19 :|: s18 >= 0, s18 <= 3 * (z - 2), s19 >= 0, s19 <= 1 + s18 + 2, z - 2 >= 0
   encArg(z) -{ 22 + s21 + -16*z + 6*z^2 }-> s22 :|: s20 >= 0, s20 <= 3 * (z - 2), s21 >= 0, s21 <= s20 + 3, s22 >= 0, s22 <= s21 + 2, z - 2 >= 0
   encArg(z) -{ 24 + s23 + s24 + -16*z + 6*z^2 }-> s25 :|: s23 >= 0, s23 <= 3 * (z - 2), s24 >= 0, s24 <= s23 + 2, s25 >= 0, s25 <= s24 + 2, z - 2 >= 0
   encArg(z) -{ 4 }-> s5 :|: s5 >= 0, s5 <= 0 + 2, z = 1 + 0
   encArg(z) -{ 4 }-> s6 :|: s6 >= 0, s6 <= 0 + 2, z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 6 + -4*z + 6*z^2 }-> 1 + s9 :|: s9 >= 0, s9 <= 3 * (z - 1), z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_a__f(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 0 + 3, z = 0
   encode_a__f(z) -{ 8 + -4*z + 6*z^2 }-> s27 :|: s26 >= 0, s26 <= 3 * (z - 1), s27 >= 0, s27 <= 1 + s26 + 3, z - 1 >= 0
   encode_a__f(z) -{ 2 }-> s3 :|: s3 >= 0, s3 <= 0 + 3, z >= 0
   encode_a__f(z) -{ 10 + -4*z + 6*z^2 }-> s30 :|: s28 >= 0, s28 <= 3 * (z - 1), s29 >= 0, s29 <= s28 + 3, s30 >= 0, s30 <= s29 + 3, z - 1 >= 0
   encode_a__f(z) -{ 12 + s31 + -4*z + 6*z^2 }-> s33 :|: s31 >= 0, s31 <= 3 * (z - 1), s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 3, z - 1 >= 0
   encode_a__f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s34 :|: s34 >= 0, s34 <= 3 * z, z >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 8 + 8*z + 6*z^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 3 * z, z >= 0
   encode_mark(z) -{ 11 + s36 + -4*z + 6*z^2 }-> s37 :|: s36 >= 0, s36 <= 3 * (z - 1), s37 >= 0, s37 <= 1 + s36 + 2, z - 1 >= 0
   encode_mark(z) -{ 12 + s39 + -4*z + 6*z^2 }-> s40 :|: s38 >= 0, s38 <= 3 * (z - 1), s39 >= 0, s39 <= s38 + 3, s40 >= 0, s40 <= s39 + 2, z - 1 >= 0
   encode_mark(z) -{ 14 + s41 + s42 + -4*z + 6*z^2 }-> s43 :|: s41 >= 0, s41 <= 3 * (z - 1), s42 >= 0, s42 <= s41 + 2, s43 >= 0, s43 <= s42 + 2, z - 1 >= 0
   encode_mark(z) -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 2, z = 0
   encode_mark(z) -{ 4 }-> s8 :|: s8 >= 0, s8 <= 0 + 2, z >= 0
   encode_mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 3 }-> s' :|: s' >= 0, s' <= z - 1 + 3, z - 1 >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 4 + z }-> 1 + s4 :|: s4 >= 0, s4 <= z - 1 + 2, z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  a__f: runtime: O(1) [2], size: O(n^1) [3 + z]
  mark: runtime: O(n^1) [4 + z], size: O(n^1) [2 + z]
  encArg: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [3*z]
  encode_mark: runtime: O(n^2) [45 + 2*z + 18*z^2], size: O(n^1) [2 + 3*z]
  encode_g: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]
  encode_f: runtime: O(n^2) [8 + 8*z + 6*z^2], size: O(n^1) [1 + 3*z]
  encode_a__f: runtime: O(n^2) [34 + 18*z^2], size: O(n^1) [3 + 3*z]

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(67) FinalProof (FINISHED)
Computed overall runtime complexity
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(68)
BOUNDS(1, n^2)

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(69) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(70)
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^2).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

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

The rewrite sequence

mark(g(X)) ->^+ g(mark(X))

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

The pumping substitution is [X / g(X)].

The result substitution is [ ].




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(72)
Complex Obligation (BEST)

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(73)
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^2).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

Rewrite Strategy: INNERMOST
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(74) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(75)
BOUNDS(n^1, INF)

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(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^2).


The TRS R consists of the following rules:

   a__f(f(a)) -> a__f(g(f(a)))
   mark(f(X)) -> a__f(X)
   mark(a) -> a
   mark(g(X)) -> g(mark(X))
   a__f(X) -> f(X)

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

   encArg(f(x_1)) -> f(encArg(x_1))
   encArg(a) -> a
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))

Rewrite Strategy: INNERMOST