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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (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), 194 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), 3 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 370 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 1138 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 1685 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 688 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 324 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) FinalProof [FINISHED, 0 ms]
    (50) BOUNDS(1, n^2)
(51) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(52) TRS for Loop Detection
    (53) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) 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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(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:

   f(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(h(x_1)) -> h(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_h(x_1) -> h(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:

   f(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(h(x_1)) -> h(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_h(x_1) -> h(encArg(x_1)) [0]

The TRS has the following type information:
f :: g:h:cons_f -> g:h:cons_f -> g:h:cons_f
g :: g:h:cons_f -> g:h:cons_f
h :: g:h:cons_f -> g:h:cons_f
encArg :: g:h:cons_f -> g:h:cons_f
cons_f :: g:h:cons_f -> g:h:cons_f -> g:h:cons_f
encode_f :: g:h:cons_f -> g:h:cons_f -> g:h:cons_f
encode_g :: g:h:cons_f -> g:h:cons_f
encode_h :: g:h:cons_f -> g:h:cons_f

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

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

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

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

(c) The following functions are completely defined:

   f_2
   encArg_1
   encode_f_2
   encode_g_1
   encode_h_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   encode_g(v0) -> const [0]
   encode_h(v0) -> const [0]
   f(v0, v1) -> const [0]

And the following fresh constants:    const

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

(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:

   f(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(h(x_1)) -> h(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_h(x_1) -> h(encArg(x_1)) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   encode_g(v0) -> const [0]
   encode_h(v0) -> const [0]
   f(v0, v1) -> const [0]

The TRS has the following type information:
f :: g:h:cons_f:const -> g:h:cons_f:const -> g:h:cons_f:const
g :: g:h:cons_f:const -> g:h:cons_f:const
h :: g:h:cons_f:const -> g:h:cons_f:const
encArg :: g:h:cons_f:const -> g:h:cons_f:const
cons_f :: g:h:cons_f:const -> g:h:cons_f:const -> g:h:cons_f:const
encode_f :: g:h:cons_f:const -> g:h:cons_f:const -> g:h:cons_f:const
encode_g :: g:h:cons_f:const -> g:h:cons_f:const
encode_h :: g:h:cons_f:const -> g:h:cons_f:const
const :: g:h:cons_f:const

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:

   f(x, g(x)) -> x [1]
   f(x, h(y)) -> f(h(x), y) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(h(x_1)) -> h(encArg(x_1)) [0]
   encArg(cons_f(g(x_1'), g(x_12))) -> f(g(encArg(x_1')), g(encArg(x_12))) [0]
   encArg(cons_f(g(x_1'), h(x_13))) -> f(g(encArg(x_1')), h(encArg(x_13))) [0]
   encArg(cons_f(g(x_1'), cons_f(x_14, x_2''))) -> f(g(encArg(x_1')), f(encArg(x_14), encArg(x_2''))) [0]
   encArg(cons_f(g(x_1'), x_2)) -> f(g(encArg(x_1')), const) [0]
   encArg(cons_f(h(x_1''), g(x_15))) -> f(h(encArg(x_1'')), g(encArg(x_15))) [0]
   encArg(cons_f(h(x_1''), h(x_16))) -> f(h(encArg(x_1'')), h(encArg(x_16))) [0]
   encArg(cons_f(h(x_1''), cons_f(x_17, x_21))) -> f(h(encArg(x_1'')), f(encArg(x_17), encArg(x_21))) [0]
   encArg(cons_f(h(x_1''), x_2)) -> f(h(encArg(x_1'')), const) [0]
   encArg(cons_f(cons_f(x_11, x_2'), g(x_18))) -> f(f(encArg(x_11), encArg(x_2')), g(encArg(x_18))) [0]
   encArg(cons_f(cons_f(x_11, x_2'), h(x_19))) -> f(f(encArg(x_11), encArg(x_2')), h(encArg(x_19))) [0]
   encArg(cons_f(cons_f(x_11, x_2'), cons_f(x_110, x_22))) -> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) [0]
   encArg(cons_f(cons_f(x_11, x_2'), x_2)) -> f(f(encArg(x_11), encArg(x_2')), const) [0]
   encArg(cons_f(x_1, g(x_111))) -> f(const, g(encArg(x_111))) [0]
   encArg(cons_f(x_1, h(x_112))) -> f(const, h(encArg(x_112))) [0]
   encArg(cons_f(x_1, cons_f(x_113, x_23))) -> f(const, f(encArg(x_113), encArg(x_23))) [0]
   encArg(cons_f(x_1, x_2)) -> f(const, const) [0]
   encode_f(g(x_114), g(x_117)) -> f(g(encArg(x_114)), g(encArg(x_117))) [0]
   encode_f(g(x_114), h(x_118)) -> f(g(encArg(x_114)), h(encArg(x_118))) [0]
   encode_f(g(x_114), cons_f(x_119, x_25)) -> f(g(encArg(x_114)), f(encArg(x_119), encArg(x_25))) [0]
   encode_f(g(x_114), x_2) -> f(g(encArg(x_114)), const) [0]
   encode_f(h(x_115), g(x_120)) -> f(h(encArg(x_115)), g(encArg(x_120))) [0]
   encode_f(h(x_115), h(x_121)) -> f(h(encArg(x_115)), h(encArg(x_121))) [0]
   encode_f(h(x_115), cons_f(x_122, x_26)) -> f(h(encArg(x_115)), f(encArg(x_122), encArg(x_26))) [0]
   encode_f(h(x_115), x_2) -> f(h(encArg(x_115)), const) [0]
   encode_f(cons_f(x_116, x_24), g(x_123)) -> f(f(encArg(x_116), encArg(x_24)), g(encArg(x_123))) [0]
   encode_f(cons_f(x_116, x_24), h(x_124)) -> f(f(encArg(x_116), encArg(x_24)), h(encArg(x_124))) [0]
   encode_f(cons_f(x_116, x_24), cons_f(x_125, x_27)) -> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) [0]
   encode_f(cons_f(x_116, x_24), x_2) -> f(f(encArg(x_116), encArg(x_24)), const) [0]
   encode_f(x_1, g(x_126)) -> f(const, g(encArg(x_126))) [0]
   encode_f(x_1, h(x_127)) -> f(const, h(encArg(x_127))) [0]
   encode_f(x_1, cons_f(x_128, x_28)) -> f(const, f(encArg(x_128), encArg(x_28))) [0]
   encode_f(x_1, x_2) -> f(const, const) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_h(x_1) -> h(encArg(x_1)) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   encode_g(v0) -> const [0]
   encode_h(v0) -> const [0]
   f(v0, v1) -> const [0]

The TRS has the following type information:
f :: g:h:cons_f:const -> g:h:cons_f:const -> g:h:cons_f:const
g :: g:h:cons_f:const -> g:h:cons_f:const
h :: g:h:cons_f:const -> g:h:cons_f:const
encArg :: g:h:cons_f:const -> g:h:cons_f:const
cons_f :: g:h:cons_f:const -> g:h:cons_f:const -> g:h:cons_f:const
encode_f :: g:h:cons_f:const -> g:h:cons_f:const -> g:h:cons_f:const
encode_g :: g:h:cons_f:const -> g:h:cons_f:const
encode_h :: g:h:cons_f:const -> g:h:cons_f:const
const :: g:h:cons_f:const

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:

   const => 0

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, x_2 >= 0, z' = x_2
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(x_123)) :|: z = 1 + x_116 + x_24, z' = 1 + x_123, x_123 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(x_124)) :|: z = 1 + x_116 + x_24, z' = 1 + x_124, x_124 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: x_1 >= 0, x_128 >= 0, z = x_1, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_126)) :|: x_1 >= 0, z' = 1 + x_126, x_126 >= 0, z = x_1
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_127)) :|: x_1 >= 0, x_127 >= 0, z' = 1 + x_127, z = x_1
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_114), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, x_114 >= 0, x_25 >= 0, z = 1 + x_114, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_114), 0) :|: x_114 >= 0, x_2 >= 0, z' = x_2, z = 1 + x_114
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_114), 1 + encArg(x_117)) :|: x_117 >= 0, x_114 >= 0, z' = 1 + x_117, z = 1 + x_114
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_114), 1 + encArg(x_118)) :|: x_114 >= 0, z' = 1 + x_118, z = 1 + x_114, x_118 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_115), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, x_115 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0, z = 1 + x_115
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_115), 0) :|: x_115 >= 0, x_2 >= 0, z = 1 + x_115, z' = x_2
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_115), 1 + encArg(x_120)) :|: x_120 >= 0, x_115 >= 0, z = 1 + x_115, z' = 1 + x_120
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_115), 1 + encArg(x_121)) :|: x_115 >= 0, z = 1 + x_115, z' = 1 + x_121, x_121 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_g(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_h(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_h(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z') -{ 1 }-> x :|: z' = 1 + x, x >= 0, z = x
   f(z, z') -{ 1 }-> f(1 + x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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(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:

   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1


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

   { f }
   { encArg }
   { encode_h }
   { encode_f }
   { encode_g }

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {f}, {encArg}, {encode_h}, {encode_f}, {encode_g}

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(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:

   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {f}, {encArg}, {encode_h}, {encode_f}, {encode_g}

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {f}, {encArg}, {encode_h}, {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: ?, size: O(n^1) [z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 }-> f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encArg}, {encode_h}, {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encArg}, {encode_h}, {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encArg}, {encode_h}, {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: ?, size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encArg
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 11*z + 6*z^2

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), f(encArg(x_110), encArg(x_22))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 0) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_18)) :|: x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 0 }-> f(f(encArg(x_11), encArg(x_2')), 1 + encArg(x_19)) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_113), encArg(x_23))) :|: x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_111)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_112)) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_2''))) :|: z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_12)) :|: z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 1 + encArg(x_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), f(encArg(x_17), encArg(x_21))) :|: x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 0) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_15)) :|: x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1''), 1 + encArg(x_16)) :|: x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), f(encArg(x_125), encArg(x_27))) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 0) :|: z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_116), encArg(x_24)), 1 + encArg(z' - 1)) :|: z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_128), encArg(x_28))) :|: z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_119), encArg(x_25))) :|: z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_122), encArg(x_26))) :|: x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_h}, {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]

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(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:

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_h}, {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_h}, {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: ?, size: O(n^1) [1 + z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_h
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 11*z + 6*z^2

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: ?, size: O(n^1) [z']

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

(41) 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: 17 + 69*z + 60*z^2 + 96*z' + 66*z'^2

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [17 + 69*z + 60*z^2 + 96*z' + 66*z'^2], size: O(n^1) [z']

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [17 + 69*z + 60*z^2 + 96*z' + 66*z'^2], size: O(n^1) [z']

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

(45) 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 + z

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [17 + 69*z + 60*z^2 + 96*z' + 66*z'^2], size: O(n^1) [z']
  encode_g: runtime: ?, size: O(n^1) [1 + z]

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(47) 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: 1 + 11*z + 6*z^2

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 5 + s10 + s11 + 11*x_1' + 6*x_1'^2 + 11*x_14 + 6*x_14^2 + 11*x_2'' + 6*x_2''^2 }-> s12 :|: s8 >= 0, s8 <= x_1', s9 >= 0, s9 <= x_14, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= s10, s12 >= 0, s12 <= s11, z = 1 + (1 + x_1') + (1 + x_14 + x_2''), x_14 >= 0, x_1' >= 0, x_2'' >= 0
   encArg(z) -{ 2 + 11*x_1' + 6*x_1'^2 }-> s14 :|: s13 >= 0, s13 <= x_1', s14 >= 0, s14 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s16 + 11*x_1'' + 6*x_1''^2 + 11*x_15 + 6*x_15^2 }-> s17 :|: s15 >= 0, s15 <= x_1'', s16 >= 0, s16 <= x_15, s17 >= 0, s17 <= 1 + s16, x_15 >= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_15)
   encArg(z) -{ 4 + s19 + 11*x_1'' + 6*x_1''^2 + 11*x_16 + 6*x_16^2 }-> s20 :|: s18 >= 0, s18 <= x_1'', s19 >= 0, s19 <= x_16, s20 >= 0, s20 <= 1 + s19, x_1'' >= 0, z = 1 + (1 + x_1'') + (1 + x_16), x_16 >= 0
   encArg(z) -{ 5 + s23 + s24 + 11*x_1'' + 6*x_1''^2 + 11*x_17 + 6*x_17^2 + 11*x_21 + 6*x_21^2 }-> s25 :|: s21 >= 0, s21 <= x_1'', s22 >= 0, s22 <= x_17, s23 >= 0, s23 <= x_21, s24 >= 0, s24 <= s23, s25 >= 0, s25 <= s24, x_1'' >= 0, x_17 >= 0, z = 1 + (1 + x_1'') + (1 + x_17 + x_21), x_21 >= 0
   encArg(z) -{ 2 + 11*x_1'' + 6*x_1''^2 }-> s27 :|: s26 >= 0, s26 <= x_1'', s27 >= 0, s27 <= 0, x_1'' >= 0, z = 1 + (1 + x_1'') + x_2, x_2 >= 0
   encArg(z) -{ 6 + s29 + s31 + 11*x_11 + 6*x_11^2 + 11*x_18 + 6*x_18^2 + 11*x_2' + 6*x_2'^2 }-> s32 :|: s28 >= 0, s28 <= x_11, s29 >= 0, s29 <= x_2', s30 >= 0, s30 <= s29, s31 >= 0, s31 <= x_18, s32 >= 0, s32 <= 1 + s31, x_11 >= 0, x_2' >= 0, x_18 >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_18)
   encArg(z) -{ 6 + s34 + s36 + 11*x_11 + 6*x_11^2 + 11*x_19 + 6*x_19^2 + 11*x_2' + 6*x_2'^2 }-> s37 :|: s33 >= 0, s33 <= x_11, s34 >= 0, s34 <= x_2', s35 >= 0, s35 <= s34, s36 >= 0, s36 <= x_19, s37 >= 0, s37 <= 1 + s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_19), x_19 >= 0
   encArg(z) -{ 4 + s3 + 11*x_1' + 6*x_1'^2 + 11*x_12 + 6*x_12^2 }-> s4 :|: s2 >= 0, s2 <= x_1', s3 >= 0, s3 <= x_12, s4 >= 0, s4 <= 1 + s3, z = 1 + (1 + x_1') + (1 + x_12), x_1' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s39 + s42 + s43 + 11*x_11 + 6*x_11^2 + 11*x_110 + 6*x_110^2 + 11*x_2' + 6*x_2'^2 + 11*x_22 + 6*x_22^2 }-> s44 :|: s38 >= 0, s38 <= x_11, s39 >= 0, s39 <= x_2', s40 >= 0, s40 <= s39, s41 >= 0, s41 <= x_110, s42 >= 0, s42 <= x_22, s43 >= 0, s43 <= s42, s44 >= 0, s44 <= s43, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + (1 + x_110 + x_22), x_110 >= 0, x_22 >= 0
   encArg(z) -{ 4 + s46 + 11*x_11 + 6*x_11^2 + 11*x_2' + 6*x_2'^2 }-> s48 :|: s45 >= 0, s45 <= x_11, s46 >= 0, s46 <= x_2', s47 >= 0, s47 <= s46, s48 >= 0, s48 <= 0, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 3 + s49 + 11*x_111 + 6*x_111^2 }-> s50 :|: s49 >= 0, s49 <= x_111, s50 >= 0, s50 <= 1 + s49, x_1 >= 0, z = 1 + x_1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 3 + s51 + 11*x_112 + 6*x_112^2 }-> s52 :|: s51 >= 0, s51 <= x_112, s52 >= 0, s52 <= 1 + s51, x_1 >= 0, z = 1 + x_1 + (1 + x_112), x_112 >= 0
   encArg(z) -{ 4 + s54 + s55 + 11*x_113 + 6*x_113^2 + 11*x_23 + 6*x_23^2 }-> s56 :|: s53 >= 0, s53 <= x_113, s54 >= 0, s54 <= x_23, s55 >= 0, s55 <= s54, s56 >= 0, s56 <= s55, x_1 >= 0, x_113 >= 0, x_23 >= 0, z = 1 + x_1 + (1 + x_113 + x_23)
   encArg(z) -{ 4 + s6 + 11*x_1' + 6*x_1'^2 + 11*x_13 + 6*x_13^2 }-> s7 :|: s5 >= 0, s5 <= x_1', s6 >= 0, s6 <= x_13, s7 >= 0, s7 <= 1 + s6, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -4 + -1*z + 6*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_f(z, z') -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ -6 + s58 + -1*z + 6*z^2 + -1*z' + 6*z'^2 }-> s59 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= z' - 1, s59 >= 0, s59 <= 1 + s58, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ s62 + s63 + 11*x_119 + 6*x_119^2 + 11*x_25 + 6*x_25^2 + -1*z + 6*z^2 }-> s64 :|: s60 >= 0, s60 <= z - 1, s61 >= 0, s61 <= x_119, s62 >= 0, s62 <= x_25, s63 >= 0, s63 <= s62, s64 >= 0, s64 <= s63, z' = 1 + x_119 + x_25, z - 1 >= 0, x_25 >= 0, x_119 >= 0
   encode_f(z, z') -{ -3 + -1*z + 6*z^2 }-> s66 :|: s65 >= 0, s65 <= z - 1, s66 >= 0, s66 <= 0, z - 1 >= 0, z' >= 0
   encode_f(z, z') -{ s69 + s70 + 11*x_122 + 6*x_122^2 + 11*x_26 + 6*x_26^2 + -1*z + 6*z^2 }-> s71 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= x_122, s69 >= 0, s69 <= x_26, s70 >= 0, s70 <= s69, s71 >= 0, s71 <= s70, x_26 >= 0, z - 1 >= 0, z' = 1 + x_122 + x_26, x_122 >= 0
   encode_f(z, z') -{ 1 + s73 + s75 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 + -1*z' + 6*z'^2 }-> s76 :|: s72 >= 0, s72 <= x_116, s73 >= 0, s73 <= x_24, s74 >= 0, s74 <= s73, s75 >= 0, s75 <= z' - 1, s76 >= 0, s76 <= 1 + s75, z = 1 + x_116 + x_24, z' - 1 >= 0, x_116 >= 0, x_24 >= 0
   encode_f(z, z') -{ 7 + s78 + s81 + s82 + 11*x_116 + 6*x_116^2 + 11*x_125 + 6*x_125^2 + 11*x_24 + 6*x_24^2 + 11*x_27 + 6*x_27^2 }-> s83 :|: s77 >= 0, s77 <= x_116, s78 >= 0, s78 <= x_24, s79 >= 0, s79 <= s78, s80 >= 0, s80 <= x_125, s81 >= 0, s81 <= x_27, s82 >= 0, s82 <= s81, s83 >= 0, s83 <= s82, z = 1 + x_116 + x_24, x_116 >= 0, x_125 >= 0, z' = 1 + x_125 + x_27, x_24 >= 0, x_27 >= 0
   encode_f(z, z') -{ 4 + s85 + 11*x_116 + 6*x_116^2 + 11*x_24 + 6*x_24^2 }-> s87 :|: s84 >= 0, s84 <= x_116, s85 >= 0, s85 <= x_24, s86 >= 0, s86 <= s85, s87 >= 0, s87 <= 0, z = 1 + x_116 + x_24, x_116 >= 0, x_24 >= 0, z' >= 0
   encode_f(z, z') -{ -2 + s88 + -1*z' + 6*z'^2 }-> s89 :|: s88 >= 0, s88 <= z' - 1, s89 >= 0, s89 <= 1 + s88, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + s92 + 11*x_128 + 6*x_128^2 + 11*x_28 + 6*x_28^2 }-> s93 :|: s90 >= 0, s90 <= x_128, s91 >= 0, s91 <= x_28, s92 >= 0, s92 <= s91, s93 >= 0, s93 <= s92, z >= 0, x_128 >= 0, z' = 1 + x_128 + x_28, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s94 :|: s94 >= 0, s94 <= z, z >= 0
   encode_h(z) -{ 0 }-> 0 :|: z >= 0
   encode_h(z) -{ 1 + 11*z + 6*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z' - 1, z >= 0, z' - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0, z = z' - 1

Function symbols to be analyzed: 
Previous analysis results are:
  f: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  encArg: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [z]
  encode_h: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]
  encode_f: runtime: O(n^2) [17 + 69*z + 60*z^2 + 96*z' + 66*z'^2], size: O(n^1) [z']
  encode_g: runtime: O(n^2) [1 + 11*z + 6*z^2], size: O(n^1) [1 + z]

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(49) FinalProof (FINISHED)
Computed overall runtime complexity
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(50)
BOUNDS(1, n^2)

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(51) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(52)
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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))

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

The rewrite sequence

f(x, h(y)) ->^+ f(h(x), y)

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

The pumping substitution is [y / h(y)].

The result substitution is [x / h(x)].




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(54)
Complex Obligation (BEST)

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(55)
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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))

Rewrite Strategy: INNERMOST
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(56) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(57)
BOUNDS(n^1, INF)

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(58)
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:

   f(x, g(x)) -> x
   f(x, h(y)) -> f(h(x), y)

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))

Rewrite Strategy: INNERMOST