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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (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), 156 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), 4 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), 3 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 70 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 560 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 1310 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 847 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 308 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) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 0 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 356 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
            (64) BOUNDS(1, INF)


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

(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, x) -> a
   f(g(x), y) -> f(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(a) -> a
   encArg(g(x_1)) -> g(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_a -> a
   encode_g(x_1) -> g(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, x) -> a
   f(g(x), y) -> f(x, y)

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

   encArg(a) -> a
   encArg(g(x_1)) -> g(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_a -> a
   encode_g(x_1) -> g(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, x) -> a
   f(g(x), y) -> f(x, y)

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

   encArg(a) -> a
   encArg(g(x_1)) -> g(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_a -> a
   encode_g(x_1) -> g(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, x) -> a [1]
   f(g(x), y) -> f(x, y) [1]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(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_a -> a [0]
   encode_g(x_1) -> g(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, x) -> a [1]
   f(g(x), y) -> f(x, y) [1]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(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_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]

The TRS has the following type information:
f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
a :: a:g:cons_f
g :: a:g:cons_f -> a:g:cons_f
encArg :: a:g:cons_f -> a:g:cons_f
cons_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_a :: a:g:cons_f
encode_g :: a:g:cons_f -> a:g: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_a
   encode_g_1

Due to the following rules being added:

   encArg(v0) -> a [0]
   encode_f(v0, v1) -> a [0]
   encode_a -> a [0]
   encode_g(v0) -> a [0]
   f(v0, v1) -> a [0]

And the following fresh constants: none

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   f(x, x) -> a [1]
   f(g(x), y) -> f(x, y) [1]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(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_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encArg(v0) -> a [0]
   encode_f(v0, v1) -> a [0]
   encode_a -> a [0]
   encode_g(v0) -> a [0]
   f(v0, v1) -> a [0]

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

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

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

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

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

   f(x, x) -> a [1]
   f(g(x), y) -> f(x, y) [1]
   encArg(a) -> a [0]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(a, a)) -> f(a, a) [0]
   encArg(cons_f(a, g(x_11))) -> f(a, g(encArg(x_11))) [0]
   encArg(cons_f(a, cons_f(x_12, x_2''))) -> f(a, f(encArg(x_12), encArg(x_2''))) [0]
   encArg(cons_f(a, x_2)) -> f(a, a) [0]
   encArg(cons_f(g(x_1'), a)) -> f(g(encArg(x_1')), a) [0]
   encArg(cons_f(g(x_1'), g(x_13))) -> f(g(encArg(x_1')), g(encArg(x_13))) [0]
   encArg(cons_f(g(x_1'), cons_f(x_14, x_21))) -> f(g(encArg(x_1')), f(encArg(x_14), encArg(x_21))) [0]
   encArg(cons_f(g(x_1'), x_2)) -> f(g(encArg(x_1')), a) [0]
   encArg(cons_f(cons_f(x_1'', x_2'), a)) -> f(f(encArg(x_1''), encArg(x_2')), a) [0]
   encArg(cons_f(cons_f(x_1'', x_2'), g(x_15))) -> f(f(encArg(x_1''), encArg(x_2')), g(encArg(x_15))) [0]
   encArg(cons_f(cons_f(x_1'', x_2'), cons_f(x_16, x_22))) -> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) [0]
   encArg(cons_f(cons_f(x_1'', x_2'), x_2)) -> f(f(encArg(x_1''), encArg(x_2')), a) [0]
   encArg(cons_f(x_1, a)) -> f(a, a) [0]
   encArg(cons_f(x_1, g(x_17))) -> f(a, g(encArg(x_17))) [0]
   encArg(cons_f(x_1, cons_f(x_18, x_23))) -> f(a, f(encArg(x_18), encArg(x_23))) [0]
   encArg(cons_f(x_1, x_2)) -> f(a, a) [0]
   encode_f(a, a) -> f(a, a) [0]
   encode_f(a, g(x_111)) -> f(a, g(encArg(x_111))) [0]
   encode_f(a, cons_f(x_112, x_25)) -> f(a, f(encArg(x_112), encArg(x_25))) [0]
   encode_f(a, x_2) -> f(a, a) [0]
   encode_f(g(x_19), a) -> f(g(encArg(x_19)), a) [0]
   encode_f(g(x_19), g(x_113)) -> f(g(encArg(x_19)), g(encArg(x_113))) [0]
   encode_f(g(x_19), cons_f(x_114, x_26)) -> f(g(encArg(x_19)), f(encArg(x_114), encArg(x_26))) [0]
   encode_f(g(x_19), x_2) -> f(g(encArg(x_19)), a) [0]
   encode_f(cons_f(x_110, x_24), a) -> f(f(encArg(x_110), encArg(x_24)), a) [0]
   encode_f(cons_f(x_110, x_24), g(x_115)) -> f(f(encArg(x_110), encArg(x_24)), g(encArg(x_115))) [0]
   encode_f(cons_f(x_110, x_24), cons_f(x_116, x_27)) -> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) [0]
   encode_f(cons_f(x_110, x_24), x_2) -> f(f(encArg(x_110), encArg(x_24)), a) [0]
   encode_f(x_1, a) -> f(a, a) [0]
   encode_f(x_1, g(x_117)) -> f(a, g(encArg(x_117))) [0]
   encode_f(x_1, cons_f(x_118, x_28)) -> f(a, f(encArg(x_118), encArg(x_28))) [0]
   encode_f(x_1, x_2) -> f(a, a) [0]
   encode_a -> a [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encArg(v0) -> a [0]
   encode_f(v0, v1) -> a [0]
   encode_a -> a [0]
   encode_g(v0) -> a [0]
   f(v0, v1) -> a [0]

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

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

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

   a => 0

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + x_1 + 0, x_1 >= 0
   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_11)) :|: x_11 >= 0, z = 1 + 0 + (1 + x_11)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), 0) :|: z = 1 + (1 + x_1') + 0, x_1' >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_2 >= 0, z' = x_2, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(x_115)) :|: x_115 >= 0, z' = 1 + x_115, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: x_1 >= 0, z = x_1, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_2 >= 0, z' = x_2, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = x_1, z' = 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_111)) :|: z' = 1 + x_111, z = 0, x_111 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 1 + encArg(x_117)) :|: x_1 >= 0, x_117 >= 0, z' = 1 + x_117, z = x_1
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z = 1 + x_19, x_19 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), 0) :|: z = 1 + x_19, z' = 0, x_19 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), 0) :|: x_2 >= 0, z' = x_2, z = 1 + x_19, x_19 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(x_19), 1 + encArg(x_113)) :|: x_113 >= 0, z = 1 + x_19, z' = 1 + x_113, x_19 >= 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
   f(z, z') -{ 1 }-> f(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   f(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { encode_a }
   { f }
   { encArg }
   { encode_f }
   { encode_g }

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(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_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(29) 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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z = 1 + 0 + 0
   encArg(z) -{ 0 }-> f(0, 0) :|: z - 1 >= 0
   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_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z = 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z' >= 0, z = 0
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' = 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 }-> f(z - 1, z') :|: z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(35) 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 + 2*z + 2*z^2

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), f(encArg(x_16), encArg(x_22))) :|: x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 0) :|: x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> f(f(encArg(x_1''), encArg(x_2')), 1 + encArg(x_15)) :|: x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_12), encArg(x_2''))) :|: z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_18), encArg(x_23))) :|: x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 0 }-> f(0, 1 + encArg(x_17)) :|: x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 0 }-> f(0, 1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(x_1'), f(encArg(x_14), encArg(x_21))) :|: x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 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_13)) :|: z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 0 }-> f(1 + encArg(z - 2), 0) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), f(encArg(x_116), encArg(x_27))) :|: x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 0) :|: z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_110), encArg(x_24)), 1 + encArg(z' - 1)) :|: z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_112), encArg(x_25))) :|: x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_118), encArg(x_28))) :|: z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, 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(0, 1 + encArg(z' - 1)) :|: z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), f(encArg(x_114), encArg(x_26))) :|: x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 0 }-> f(1 + encArg(z - 1), 0) :|: z' >= 0, z - 1 >= 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
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 4 + s9 + 2*x_12 + 2*x_12^2 + 2*x_2'' + 2*x_2''^2 }-> s12 :|: s9 >= 0, s9 <= x_12, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= 0, s12 >= 0, s12 <= 0, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s13 + -6*z + 2*z^2 }-> s14 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 0, z - 2 >= 0
   encArg(z) -{ 4 + s15 + 2*x_1' + 2*x_1'^2 + 2*x_13 + 2*x_13^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_13, s17 >= 0, s17 <= 0, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 6 + s18 + s19 + 2*x_1' + 2*x_1'^2 + 2*x_14 + 2*x_14^2 + 2*x_21 + 2*x_21^2 }-> s22 :|: s18 >= 0, s18 <= x_1', s19 >= 0, s19 <= x_14, s20 >= 0, s20 <= x_21, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s23 + 2*x_1' + 2*x_1'^2 }-> s24 :|: s23 >= 0, s23 <= x_1', s24 >= 0, s24 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s25 + s27 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s28 :|: s25 >= 0, s25 <= x_1'', s26 >= 0, s26 <= x_2', s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s29 + s31 + 2*x_1'' + 2*x_1''^2 + 2*x_15 + 2*x_15^2 + 2*x_2' + 2*x_2'^2 }-> s33 :|: s29 >= 0, s29 <= x_1'', s30 >= 0, s30 <= x_2', s31 >= 0, s31 <= 0, s32 >= 0, s32 <= x_15, s33 >= 0, s33 <= 0, x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s34 + s36 + s37 + 2*x_1'' + 2*x_1''^2 + 2*x_16 + 2*x_16^2 + 2*x_2' + 2*x_2'^2 + 2*x_22 + 2*x_22^2 }-> s40 :|: s34 >= 0, s34 <= x_1'', s35 >= 0, s35 <= x_2', s36 >= 0, s36 <= 0, s37 >= 0, s37 <= x_16, s38 >= 0, s38 <= x_22, s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s41 + s43 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s44 :|: s41 >= 0, s41 <= x_1'', s42 >= 0, s42 <= x_2', s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 2*x_17 + 2*x_17^2 }-> s46 :|: s45 >= 0, s45 <= x_17, s46 >= 0, s46 <= 0, x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s47 + 2*x_18 + 2*x_18^2 + 2*x_23 + 2*x_23^2 }-> s50 :|: s47 >= 0, s47 <= x_18, s48 >= 0, s48 <= x_23, s49 >= 0, s49 <= 0, s50 >= 0, s50 <= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 6 + -6*z + 2*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 + -2*z + 2*z^2 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s52 :|: s51 >= 0, s51 <= z' - 1, s52 >= 0, s52 <= 0, z = 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s53 + 2*x_112 + 2*x_112^2 + 2*x_25 + 2*x_25^2 }-> s56 :|: s53 >= 0, s53 <= x_112, s54 >= 0, s54 <= x_25, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 3 + s57 + -2*z + 2*z^2 }-> s58 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= 0, z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s59 + -2*z + 2*z^2 + -2*z' + 2*z'^2 }-> s61 :|: s59 >= 0, s59 <= z - 1, s60 >= 0, s60 <= z' - 1, s61 >= 0, s61 <= 0, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 6 + s62 + s63 + 2*x_114 + 2*x_114^2 + 2*x_26 + 2*x_26^2 + -2*z + 2*z^2 }-> s66 :|: s62 >= 0, s62 <= z - 1, s63 >= 0, s63 <= x_114, s64 >= 0, s64 <= x_26, s65 >= 0, s65 <= 0, s66 >= 0, s66 <= 0, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 3 + s67 + -2*z + 2*z^2 }-> s68 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= 0, z' >= 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s69 + s71 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s72 :|: s69 >= 0, s69 <= x_110, s70 >= 0, s70 <= x_24, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 5 + s73 + s75 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 + -2*z' + 2*z'^2 }-> s77 :|: s73 >= 0, s73 <= x_110, s74 >= 0, s74 <= x_24, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= z' - 1, s77 >= 0, s77 <= 0, z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 7 + s78 + s80 + s81 + 2*x_110 + 2*x_110^2 + 2*x_116 + 2*x_116^2 + 2*x_24 + 2*x_24^2 + 2*x_27 + 2*x_27^2 }-> s84 :|: s78 >= 0, s78 <= x_110, s79 >= 0, s79 <= x_24, s80 >= 0, s80 <= 0, s81 >= 0, s81 <= x_116, s82 >= 0, s82 <= x_27, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 4 + s85 + s87 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s88 :|: s85 >= 0, s85 <= x_110, s86 >= 0, s86 <= x_24, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s90 :|: s89 >= 0, s89 <= z' - 1, s90 >= 0, s90 <= 0, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + 2*x_118 + 2*x_118^2 + 2*x_28 + 2*x_28^2 }-> s94 :|: s91 >= 0, s91 <= x_118, s92 >= 0, s92 <= x_28, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 2*z + 2*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}, {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z], size: O(1) [0]
  encArg: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(n^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(1) with polynomial bound: 0

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 4 + s9 + 2*x_12 + 2*x_12^2 + 2*x_2'' + 2*x_2''^2 }-> s12 :|: s9 >= 0, s9 <= x_12, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= 0, s12 >= 0, s12 <= 0, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s13 + -6*z + 2*z^2 }-> s14 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 0, z - 2 >= 0
   encArg(z) -{ 4 + s15 + 2*x_1' + 2*x_1'^2 + 2*x_13 + 2*x_13^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_13, s17 >= 0, s17 <= 0, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 6 + s18 + s19 + 2*x_1' + 2*x_1'^2 + 2*x_14 + 2*x_14^2 + 2*x_21 + 2*x_21^2 }-> s22 :|: s18 >= 0, s18 <= x_1', s19 >= 0, s19 <= x_14, s20 >= 0, s20 <= x_21, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s23 + 2*x_1' + 2*x_1'^2 }-> s24 :|: s23 >= 0, s23 <= x_1', s24 >= 0, s24 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s25 + s27 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s28 :|: s25 >= 0, s25 <= x_1'', s26 >= 0, s26 <= x_2', s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s29 + s31 + 2*x_1'' + 2*x_1''^2 + 2*x_15 + 2*x_15^2 + 2*x_2' + 2*x_2'^2 }-> s33 :|: s29 >= 0, s29 <= x_1'', s30 >= 0, s30 <= x_2', s31 >= 0, s31 <= 0, s32 >= 0, s32 <= x_15, s33 >= 0, s33 <= 0, x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s34 + s36 + s37 + 2*x_1'' + 2*x_1''^2 + 2*x_16 + 2*x_16^2 + 2*x_2' + 2*x_2'^2 + 2*x_22 + 2*x_22^2 }-> s40 :|: s34 >= 0, s34 <= x_1'', s35 >= 0, s35 <= x_2', s36 >= 0, s36 <= 0, s37 >= 0, s37 <= x_16, s38 >= 0, s38 <= x_22, s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s41 + s43 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s44 :|: s41 >= 0, s41 <= x_1'', s42 >= 0, s42 <= x_2', s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 2*x_17 + 2*x_17^2 }-> s46 :|: s45 >= 0, s45 <= x_17, s46 >= 0, s46 <= 0, x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s47 + 2*x_18 + 2*x_18^2 + 2*x_23 + 2*x_23^2 }-> s50 :|: s47 >= 0, s47 <= x_18, s48 >= 0, s48 <= x_23, s49 >= 0, s49 <= 0, s50 >= 0, s50 <= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 6 + -6*z + 2*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 + -2*z + 2*z^2 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s52 :|: s51 >= 0, s51 <= z' - 1, s52 >= 0, s52 <= 0, z = 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s53 + 2*x_112 + 2*x_112^2 + 2*x_25 + 2*x_25^2 }-> s56 :|: s53 >= 0, s53 <= x_112, s54 >= 0, s54 <= x_25, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 3 + s57 + -2*z + 2*z^2 }-> s58 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= 0, z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s59 + -2*z + 2*z^2 + -2*z' + 2*z'^2 }-> s61 :|: s59 >= 0, s59 <= z - 1, s60 >= 0, s60 <= z' - 1, s61 >= 0, s61 <= 0, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 6 + s62 + s63 + 2*x_114 + 2*x_114^2 + 2*x_26 + 2*x_26^2 + -2*z + 2*z^2 }-> s66 :|: s62 >= 0, s62 <= z - 1, s63 >= 0, s63 <= x_114, s64 >= 0, s64 <= x_26, s65 >= 0, s65 <= 0, s66 >= 0, s66 <= 0, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 3 + s67 + -2*z + 2*z^2 }-> s68 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= 0, z' >= 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s69 + s71 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s72 :|: s69 >= 0, s69 <= x_110, s70 >= 0, s70 <= x_24, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 5 + s73 + s75 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 + -2*z' + 2*z'^2 }-> s77 :|: s73 >= 0, s73 <= x_110, s74 >= 0, s74 <= x_24, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= z' - 1, s77 >= 0, s77 <= 0, z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 7 + s78 + s80 + s81 + 2*x_110 + 2*x_110^2 + 2*x_116 + 2*x_116^2 + 2*x_24 + 2*x_24^2 + 2*x_27 + 2*x_27^2 }-> s84 :|: s78 >= 0, s78 <= x_110, s79 >= 0, s79 <= x_24, s80 >= 0, s80 <= 0, s81 >= 0, s81 <= x_116, s82 >= 0, s82 <= x_27, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 4 + s85 + s87 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s88 :|: s85 >= 0, s85 <= x_110, s86 >= 0, s86 <= x_24, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s90 :|: s89 >= 0, s89 <= z' - 1, s90 >= 0, s90 <= 0, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + 2*x_118 + 2*x_118^2 + 2*x_28 + 2*x_28^2 }-> s94 :|: s91 >= 0, s91 <= x_118, s92 >= 0, s92 <= x_28, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 2*z + 2*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(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: 52 + 20*z + 24*z^2 + 20*z' + 24*z'^2

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 4 + s9 + 2*x_12 + 2*x_12^2 + 2*x_2'' + 2*x_2''^2 }-> s12 :|: s9 >= 0, s9 <= x_12, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= 0, s12 >= 0, s12 <= 0, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s13 + -6*z + 2*z^2 }-> s14 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 0, z - 2 >= 0
   encArg(z) -{ 4 + s15 + 2*x_1' + 2*x_1'^2 + 2*x_13 + 2*x_13^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_13, s17 >= 0, s17 <= 0, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 6 + s18 + s19 + 2*x_1' + 2*x_1'^2 + 2*x_14 + 2*x_14^2 + 2*x_21 + 2*x_21^2 }-> s22 :|: s18 >= 0, s18 <= x_1', s19 >= 0, s19 <= x_14, s20 >= 0, s20 <= x_21, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s23 + 2*x_1' + 2*x_1'^2 }-> s24 :|: s23 >= 0, s23 <= x_1', s24 >= 0, s24 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s25 + s27 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s28 :|: s25 >= 0, s25 <= x_1'', s26 >= 0, s26 <= x_2', s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s29 + s31 + 2*x_1'' + 2*x_1''^2 + 2*x_15 + 2*x_15^2 + 2*x_2' + 2*x_2'^2 }-> s33 :|: s29 >= 0, s29 <= x_1'', s30 >= 0, s30 <= x_2', s31 >= 0, s31 <= 0, s32 >= 0, s32 <= x_15, s33 >= 0, s33 <= 0, x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s34 + s36 + s37 + 2*x_1'' + 2*x_1''^2 + 2*x_16 + 2*x_16^2 + 2*x_2' + 2*x_2'^2 + 2*x_22 + 2*x_22^2 }-> s40 :|: s34 >= 0, s34 <= x_1'', s35 >= 0, s35 <= x_2', s36 >= 0, s36 <= 0, s37 >= 0, s37 <= x_16, s38 >= 0, s38 <= x_22, s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s41 + s43 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s44 :|: s41 >= 0, s41 <= x_1'', s42 >= 0, s42 <= x_2', s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 2*x_17 + 2*x_17^2 }-> s46 :|: s45 >= 0, s45 <= x_17, s46 >= 0, s46 <= 0, x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s47 + 2*x_18 + 2*x_18^2 + 2*x_23 + 2*x_23^2 }-> s50 :|: s47 >= 0, s47 <= x_18, s48 >= 0, s48 <= x_23, s49 >= 0, s49 <= 0, s50 >= 0, s50 <= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 6 + -6*z + 2*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 + -2*z + 2*z^2 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s52 :|: s51 >= 0, s51 <= z' - 1, s52 >= 0, s52 <= 0, z = 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s53 + 2*x_112 + 2*x_112^2 + 2*x_25 + 2*x_25^2 }-> s56 :|: s53 >= 0, s53 <= x_112, s54 >= 0, s54 <= x_25, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 3 + s57 + -2*z + 2*z^2 }-> s58 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= 0, z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s59 + -2*z + 2*z^2 + -2*z' + 2*z'^2 }-> s61 :|: s59 >= 0, s59 <= z - 1, s60 >= 0, s60 <= z' - 1, s61 >= 0, s61 <= 0, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 6 + s62 + s63 + 2*x_114 + 2*x_114^2 + 2*x_26 + 2*x_26^2 + -2*z + 2*z^2 }-> s66 :|: s62 >= 0, s62 <= z - 1, s63 >= 0, s63 <= x_114, s64 >= 0, s64 <= x_26, s65 >= 0, s65 <= 0, s66 >= 0, s66 <= 0, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 3 + s67 + -2*z + 2*z^2 }-> s68 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= 0, z' >= 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s69 + s71 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s72 :|: s69 >= 0, s69 <= x_110, s70 >= 0, s70 <= x_24, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 5 + s73 + s75 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 + -2*z' + 2*z'^2 }-> s77 :|: s73 >= 0, s73 <= x_110, s74 >= 0, s74 <= x_24, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= z' - 1, s77 >= 0, s77 <= 0, z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 7 + s78 + s80 + s81 + 2*x_110 + 2*x_110^2 + 2*x_116 + 2*x_116^2 + 2*x_24 + 2*x_24^2 + 2*x_27 + 2*x_27^2 }-> s84 :|: s78 >= 0, s78 <= x_110, s79 >= 0, s79 <= x_24, s80 >= 0, s80 <= 0, s81 >= 0, s81 <= x_116, s82 >= 0, s82 <= x_27, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 4 + s85 + s87 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s88 :|: s85 >= 0, s85 <= x_110, s86 >= 0, s86 <= x_24, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s90 :|: s89 >= 0, s89 <= z' - 1, s90 >= 0, s90 <= 0, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + 2*x_118 + 2*x_118^2 + 2*x_28 + 2*x_28^2 }-> s94 :|: s91 >= 0, s91 <= x_118, s92 >= 0, s92 <= x_28, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 2*z + 2*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z], size: O(1) [0]
  encArg: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [52 + 20*z + 24*z^2 + 20*z' + 24*z'^2], size: O(1) [0]

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

(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, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 4 + s9 + 2*x_12 + 2*x_12^2 + 2*x_2'' + 2*x_2''^2 }-> s12 :|: s9 >= 0, s9 <= x_12, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= 0, s12 >= 0, s12 <= 0, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s13 + -6*z + 2*z^2 }-> s14 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 0, z - 2 >= 0
   encArg(z) -{ 4 + s15 + 2*x_1' + 2*x_1'^2 + 2*x_13 + 2*x_13^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_13, s17 >= 0, s17 <= 0, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 6 + s18 + s19 + 2*x_1' + 2*x_1'^2 + 2*x_14 + 2*x_14^2 + 2*x_21 + 2*x_21^2 }-> s22 :|: s18 >= 0, s18 <= x_1', s19 >= 0, s19 <= x_14, s20 >= 0, s20 <= x_21, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s23 + 2*x_1' + 2*x_1'^2 }-> s24 :|: s23 >= 0, s23 <= x_1', s24 >= 0, s24 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s25 + s27 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s28 :|: s25 >= 0, s25 <= x_1'', s26 >= 0, s26 <= x_2', s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s29 + s31 + 2*x_1'' + 2*x_1''^2 + 2*x_15 + 2*x_15^2 + 2*x_2' + 2*x_2'^2 }-> s33 :|: s29 >= 0, s29 <= x_1'', s30 >= 0, s30 <= x_2', s31 >= 0, s31 <= 0, s32 >= 0, s32 <= x_15, s33 >= 0, s33 <= 0, x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s34 + s36 + s37 + 2*x_1'' + 2*x_1''^2 + 2*x_16 + 2*x_16^2 + 2*x_2' + 2*x_2'^2 + 2*x_22 + 2*x_22^2 }-> s40 :|: s34 >= 0, s34 <= x_1'', s35 >= 0, s35 <= x_2', s36 >= 0, s36 <= 0, s37 >= 0, s37 <= x_16, s38 >= 0, s38 <= x_22, s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s41 + s43 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s44 :|: s41 >= 0, s41 <= x_1'', s42 >= 0, s42 <= x_2', s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 2*x_17 + 2*x_17^2 }-> s46 :|: s45 >= 0, s45 <= x_17, s46 >= 0, s46 <= 0, x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s47 + 2*x_18 + 2*x_18^2 + 2*x_23 + 2*x_23^2 }-> s50 :|: s47 >= 0, s47 <= x_18, s48 >= 0, s48 <= x_23, s49 >= 0, s49 <= 0, s50 >= 0, s50 <= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 6 + -6*z + 2*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 + -2*z + 2*z^2 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s52 :|: s51 >= 0, s51 <= z' - 1, s52 >= 0, s52 <= 0, z = 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s53 + 2*x_112 + 2*x_112^2 + 2*x_25 + 2*x_25^2 }-> s56 :|: s53 >= 0, s53 <= x_112, s54 >= 0, s54 <= x_25, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 3 + s57 + -2*z + 2*z^2 }-> s58 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= 0, z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s59 + -2*z + 2*z^2 + -2*z' + 2*z'^2 }-> s61 :|: s59 >= 0, s59 <= z - 1, s60 >= 0, s60 <= z' - 1, s61 >= 0, s61 <= 0, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 6 + s62 + s63 + 2*x_114 + 2*x_114^2 + 2*x_26 + 2*x_26^2 + -2*z + 2*z^2 }-> s66 :|: s62 >= 0, s62 <= z - 1, s63 >= 0, s63 <= x_114, s64 >= 0, s64 <= x_26, s65 >= 0, s65 <= 0, s66 >= 0, s66 <= 0, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 3 + s67 + -2*z + 2*z^2 }-> s68 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= 0, z' >= 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s69 + s71 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s72 :|: s69 >= 0, s69 <= x_110, s70 >= 0, s70 <= x_24, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 5 + s73 + s75 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 + -2*z' + 2*z'^2 }-> s77 :|: s73 >= 0, s73 <= x_110, s74 >= 0, s74 <= x_24, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= z' - 1, s77 >= 0, s77 <= 0, z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 7 + s78 + s80 + s81 + 2*x_110 + 2*x_110^2 + 2*x_116 + 2*x_116^2 + 2*x_24 + 2*x_24^2 + 2*x_27 + 2*x_27^2 }-> s84 :|: s78 >= 0, s78 <= x_110, s79 >= 0, s79 <= x_24, s80 >= 0, s80 <= 0, s81 >= 0, s81 <= x_116, s82 >= 0, s82 <= x_27, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 4 + s85 + s87 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s88 :|: s85 >= 0, s85 <= x_110, s86 >= 0, s86 <= x_24, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s90 :|: s89 >= 0, s89 <= z' - 1, s90 >= 0, s90 <= 0, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + 2*x_118 + 2*x_118^2 + 2*x_28 + 2*x_28^2 }-> s94 :|: s91 >= 0, s91 <= x_118, s92 >= 0, s92 <= x_28, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 2*z + 2*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z], size: O(1) [0]
  encArg: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [52 + 20*z + 24*z^2 + 20*z' + 24*z'^2], size: O(1) [0]

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

(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, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 4 + s9 + 2*x_12 + 2*x_12^2 + 2*x_2'' + 2*x_2''^2 }-> s12 :|: s9 >= 0, s9 <= x_12, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= 0, s12 >= 0, s12 <= 0, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s13 + -6*z + 2*z^2 }-> s14 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 0, z - 2 >= 0
   encArg(z) -{ 4 + s15 + 2*x_1' + 2*x_1'^2 + 2*x_13 + 2*x_13^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_13, s17 >= 0, s17 <= 0, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 6 + s18 + s19 + 2*x_1' + 2*x_1'^2 + 2*x_14 + 2*x_14^2 + 2*x_21 + 2*x_21^2 }-> s22 :|: s18 >= 0, s18 <= x_1', s19 >= 0, s19 <= x_14, s20 >= 0, s20 <= x_21, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s23 + 2*x_1' + 2*x_1'^2 }-> s24 :|: s23 >= 0, s23 <= x_1', s24 >= 0, s24 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s25 + s27 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s28 :|: s25 >= 0, s25 <= x_1'', s26 >= 0, s26 <= x_2', s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s29 + s31 + 2*x_1'' + 2*x_1''^2 + 2*x_15 + 2*x_15^2 + 2*x_2' + 2*x_2'^2 }-> s33 :|: s29 >= 0, s29 <= x_1'', s30 >= 0, s30 <= x_2', s31 >= 0, s31 <= 0, s32 >= 0, s32 <= x_15, s33 >= 0, s33 <= 0, x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s34 + s36 + s37 + 2*x_1'' + 2*x_1''^2 + 2*x_16 + 2*x_16^2 + 2*x_2' + 2*x_2'^2 + 2*x_22 + 2*x_22^2 }-> s40 :|: s34 >= 0, s34 <= x_1'', s35 >= 0, s35 <= x_2', s36 >= 0, s36 <= 0, s37 >= 0, s37 <= x_16, s38 >= 0, s38 <= x_22, s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s41 + s43 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s44 :|: s41 >= 0, s41 <= x_1'', s42 >= 0, s42 <= x_2', s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 2*x_17 + 2*x_17^2 }-> s46 :|: s45 >= 0, s45 <= x_17, s46 >= 0, s46 <= 0, x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s47 + 2*x_18 + 2*x_18^2 + 2*x_23 + 2*x_23^2 }-> s50 :|: s47 >= 0, s47 <= x_18, s48 >= 0, s48 <= x_23, s49 >= 0, s49 <= 0, s50 >= 0, s50 <= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 6 + -6*z + 2*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 + -2*z + 2*z^2 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s52 :|: s51 >= 0, s51 <= z' - 1, s52 >= 0, s52 <= 0, z = 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s53 + 2*x_112 + 2*x_112^2 + 2*x_25 + 2*x_25^2 }-> s56 :|: s53 >= 0, s53 <= x_112, s54 >= 0, s54 <= x_25, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 3 + s57 + -2*z + 2*z^2 }-> s58 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= 0, z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s59 + -2*z + 2*z^2 + -2*z' + 2*z'^2 }-> s61 :|: s59 >= 0, s59 <= z - 1, s60 >= 0, s60 <= z' - 1, s61 >= 0, s61 <= 0, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 6 + s62 + s63 + 2*x_114 + 2*x_114^2 + 2*x_26 + 2*x_26^2 + -2*z + 2*z^2 }-> s66 :|: s62 >= 0, s62 <= z - 1, s63 >= 0, s63 <= x_114, s64 >= 0, s64 <= x_26, s65 >= 0, s65 <= 0, s66 >= 0, s66 <= 0, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 3 + s67 + -2*z + 2*z^2 }-> s68 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= 0, z' >= 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s69 + s71 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s72 :|: s69 >= 0, s69 <= x_110, s70 >= 0, s70 <= x_24, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 5 + s73 + s75 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 + -2*z' + 2*z'^2 }-> s77 :|: s73 >= 0, s73 <= x_110, s74 >= 0, s74 <= x_24, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= z' - 1, s77 >= 0, s77 <= 0, z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 7 + s78 + s80 + s81 + 2*x_110 + 2*x_110^2 + 2*x_116 + 2*x_116^2 + 2*x_24 + 2*x_24^2 + 2*x_27 + 2*x_27^2 }-> s84 :|: s78 >= 0, s78 <= x_110, s79 >= 0, s79 <= x_24, s80 >= 0, s80 <= 0, s81 >= 0, s81 <= x_116, s82 >= 0, s82 <= x_27, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 4 + s85 + s87 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s88 :|: s85 >= 0, s85 <= x_110, s86 >= 0, s86 <= x_24, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s90 :|: s89 >= 0, s89 <= z' - 1, s90 >= 0, s90 <= 0, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + 2*x_118 + 2*x_118^2 + 2*x_28 + 2*x_28^2 }-> s94 :|: s91 >= 0, s91 <= x_118, s92 >= 0, s92 <= x_28, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 2*z + 2*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_g}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z], size: O(1) [0]
  encArg: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [52 + 20*z + 24*z^2 + 20*z' + 24*z'^2], size: O(1) [0]
  encode_g: runtime: ?, size: O(n^1) [1 + z]

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

(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 + 2*z + 2*z^2

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

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

   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z = 1 + 0 + 0
   encArg(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   encArg(z) -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 4 + s9 + 2*x_12 + 2*x_12^2 + 2*x_2'' + 2*x_2''^2 }-> s12 :|: s9 >= 0, s9 <= x_12, s10 >= 0, s10 <= x_2'', s11 >= 0, s11 <= 0, s12 >= 0, s12 <= 0, z = 1 + 0 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0
   encArg(z) -{ 7 + s13 + -6*z + 2*z^2 }-> s14 :|: s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 0, z - 2 >= 0
   encArg(z) -{ 4 + s15 + 2*x_1' + 2*x_1'^2 + 2*x_13 + 2*x_13^2 }-> s17 :|: s15 >= 0, s15 <= x_1', s16 >= 0, s16 <= x_13, s17 >= 0, s17 <= 0, z = 1 + (1 + x_1') + (1 + x_13), x_13 >= 0, x_1' >= 0
   encArg(z) -{ 6 + s18 + s19 + 2*x_1' + 2*x_1'^2 + 2*x_14 + 2*x_14^2 + 2*x_21 + 2*x_21^2 }-> s22 :|: s18 >= 0, s18 <= x_1', s19 >= 0, s19 <= x_14, s20 >= 0, s20 <= x_21, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, x_14 >= 0, z = 1 + (1 + x_1') + (1 + x_14 + x_21), x_1' >= 0, x_21 >= 0
   encArg(z) -{ 3 + s23 + 2*x_1' + 2*x_1'^2 }-> s24 :|: s23 >= 0, s23 <= x_1', s24 >= 0, s24 <= 0, x_1' >= 0, x_2 >= 0, z = 1 + (1 + x_1') + x_2
   encArg(z) -{ 4 + s25 + s27 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s28 :|: s25 >= 0, s25 <= x_1'', s26 >= 0, s26 <= x_2', s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + 0
   encArg(z) -{ 5 + s29 + s31 + 2*x_1'' + 2*x_1''^2 + 2*x_15 + 2*x_15^2 + 2*x_2' + 2*x_2'^2 }-> s33 :|: s29 >= 0, s29 <= x_1'', s30 >= 0, s30 <= x_2', s31 >= 0, s31 <= 0, s32 >= 0, s32 <= x_15, s33 >= 0, s33 <= 0, x_15 >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_15), x_1'' >= 0, x_2' >= 0
   encArg(z) -{ 7 + s34 + s36 + s37 + 2*x_1'' + 2*x_1''^2 + 2*x_16 + 2*x_16^2 + 2*x_2' + 2*x_2'^2 + 2*x_22 + 2*x_22^2 }-> s40 :|: s34 >= 0, s34 <= x_1'', s35 >= 0, s35 <= x_2', s36 >= 0, s36 <= 0, s37 >= 0, s37 <= x_16, s38 >= 0, s38 <= x_22, s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, x_1'' >= 0, x_16 >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + (1 + x_16 + x_22), x_22 >= 0
   encArg(z) -{ 4 + s41 + s43 + 2*x_1'' + 2*x_1''^2 + 2*x_2' + 2*x_2'^2 }-> s44 :|: s41 >= 0, s41 <= x_1'', s42 >= 0, s42 <= x_2', s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, x_1'' >= 0, x_2' >= 0, z = 1 + (1 + x_1'' + x_2') + x_2, x_2 >= 0
   encArg(z) -{ 2 + 2*x_17 + 2*x_17^2 }-> s46 :|: s45 >= 0, s45 <= x_17, s46 >= 0, s46 <= 0, x_1 >= 0, x_17 >= 0, z = 1 + x_1 + (1 + x_17)
   encArg(z) -{ 4 + s47 + 2*x_18 + 2*x_18^2 + 2*x_23 + 2*x_23^2 }-> s50 :|: s47 >= 0, s47 <= x_18, s48 >= 0, s48 <= x_23, s49 >= 0, s49 <= 0, s50 >= 0, s50 <= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_18 + x_23), x_23 >= 0, x_18 >= 0
   encArg(z) -{ 6 + -6*z + 2*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 1 + -2*z + 2*z^2 }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0, z = 0, z' = 0
   encode_f(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z = 0
   encode_f(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0, z' = 0
   encode_f(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s52 :|: s51 >= 0, s51 <= z' - 1, s52 >= 0, s52 <= 0, z = 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s53 + 2*x_112 + 2*x_112^2 + 2*x_25 + 2*x_25^2 }-> s56 :|: s53 >= 0, s53 <= x_112, s54 >= 0, s54 <= x_25, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, x_25 >= 0, z' = 1 + x_112 + x_25, z = 0, x_112 >= 0
   encode_f(z, z') -{ 3 + s57 + -2*z + 2*z^2 }-> s58 :|: s57 >= 0, s57 <= z - 1, s58 >= 0, s58 <= 0, z' = 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s59 + -2*z + 2*z^2 + -2*z' + 2*z'^2 }-> s61 :|: s59 >= 0, s59 <= z - 1, s60 >= 0, s60 <= z' - 1, s61 >= 0, s61 <= 0, z' - 1 >= 0, z - 1 >= 0
   encode_f(z, z') -{ 6 + s62 + s63 + 2*x_114 + 2*x_114^2 + 2*x_26 + 2*x_26^2 + -2*z + 2*z^2 }-> s66 :|: s62 >= 0, s62 <= z - 1, s63 >= 0, s63 <= x_114, s64 >= 0, s64 <= x_26, s65 >= 0, s65 <= 0, s66 >= 0, s66 <= 0, x_114 >= 0, x_26 >= 0, z' = 1 + x_114 + x_26, z - 1 >= 0
   encode_f(z, z') -{ 3 + s67 + -2*z + 2*z^2 }-> s68 :|: s67 >= 0, s67 <= z - 1, s68 >= 0, s68 <= 0, z' >= 0, z - 1 >= 0
   encode_f(z, z') -{ 4 + s69 + s71 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s72 :|: s69 >= 0, s69 <= x_110, s70 >= 0, s70 <= x_24, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0, z' = 0
   encode_f(z, z') -{ 5 + s73 + s75 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 + -2*z' + 2*z'^2 }-> s77 :|: s73 >= 0, s73 <= x_110, s74 >= 0, s74 <= x_24, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= z' - 1, s77 >= 0, s77 <= 0, z' - 1 >= 0, z = 1 + x_110 + x_24, x_24 >= 0, x_110 >= 0
   encode_f(z, z') -{ 7 + s78 + s80 + s81 + 2*x_110 + 2*x_110^2 + 2*x_116 + 2*x_116^2 + 2*x_24 + 2*x_24^2 + 2*x_27 + 2*x_27^2 }-> s84 :|: s78 >= 0, s78 <= x_110, s79 >= 0, s79 <= x_24, s80 >= 0, s80 <= 0, s81 >= 0, s81 <= x_116, s82 >= 0, s82 <= x_27, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, x_116 >= 0, z' = 1 + x_116 + x_27, z = 1 + x_110 + x_24, x_24 >= 0, x_27 >= 0, x_110 >= 0
   encode_f(z, z') -{ 4 + s85 + s87 + 2*x_110 + 2*x_110^2 + 2*x_24 + 2*x_24^2 }-> s88 :|: s85 >= 0, s85 <= x_110, s86 >= 0, s86 <= x_24, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, z = 1 + x_110 + x_24, x_24 >= 0, z' >= 0, x_110 >= 0
   encode_f(z, z') -{ 2 + -2*z' + 2*z'^2 }-> s90 :|: s89 >= 0, s89 <= z' - 1, s90 >= 0, s90 <= 0, z >= 0, z' - 1 >= 0
   encode_f(z, z') -{ 4 + s91 + 2*x_118 + 2*x_118^2 + 2*x_28 + 2*x_28^2 }-> s94 :|: s91 >= 0, s91 <= x_118, s92 >= 0, s92 <= x_28, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, z >= 0, z' = 1 + x_118 + x_28, x_118 >= 0, x_28 >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 1 + 2*z + 2*z^2 }-> 1 + s95 :|: s95 >= 0, s95 <= z, z >= 0
   f(z, z') -{ 1 + z }-> s :|: s >= 0, s <= 0, z - 1 >= 0, z' >= 0
   f(z, z') -{ 1 }-> 0 :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  f: runtime: O(n^1) [1 + z], size: O(1) [0]
  encArg: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(n^1) [z]
  encode_f: runtime: O(n^2) [52 + 20*z + 24*z^2 + 20*z' + 24*z'^2], size: O(1) [0]
  encode_g: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(n^1) [1 + z]

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

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

(50)
BOUNDS(1, n^2)

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

(51) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

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

   encArg(a) -> a
   encArg(g(x_1)) -> g(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_a -> a
   encode_g(x_1) -> g(encArg(x_1))

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

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

(54)
Obligation:
Innermost TRS:
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
encArg(a) -> a
encArg(g(x_1)) -> g(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_a -> a
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
a :: a:g:cons_f
g :: a:g:cons_f -> a:g:cons_f
encArg :: a:g:cons_f -> a:g:cons_f
cons_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_a :: a:g:cons_f
encode_g :: a:g:cons_f -> a:g:cons_f
hole_a:g:cons_f1_0 :: a:g:cons_f
gen_a:g:cons_f2_0 :: Nat -> a:g:cons_f

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, encArg

They will be analysed ascendingly in the following order:
f < encArg

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

(56)
Obligation:
Innermost TRS:
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
encArg(a) -> a
encArg(g(x_1)) -> g(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_a -> a
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
a :: a:g:cons_f
g :: a:g:cons_f -> a:g:cons_f
encArg :: a:g:cons_f -> a:g:cons_f
cons_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_a :: a:g:cons_f
encode_g :: a:g:cons_f -> a:g:cons_f
hole_a:g:cons_f1_0 :: a:g:cons_f
gen_a:g:cons_f2_0 :: Nat -> a:g:cons_f


Generator Equations:
gen_a:g:cons_f2_0(0) <=> a
gen_a:g:cons_f2_0(+(x, 1)) <=> g(gen_a:g:cons_f2_0(x))


The following defined symbols remain to be analysed:
f, encArg

They will be analysed ascendingly in the following order:
f < encArg

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_a:g:cons_f2_0(+(1, n4_0)), gen_a:g:cons_f2_0(b)) -> *3_0, rt in Omega(n4_0)

Induction Base:
f(gen_a:g:cons_f2_0(+(1, 0)), gen_a:g:cons_f2_0(b))

Induction Step:
f(gen_a:g:cons_f2_0(+(1, +(n4_0, 1))), gen_a:g:cons_f2_0(b)) ->_R^Omega(1)
f(gen_a:g:cons_f2_0(+(1, n4_0)), gen_a:g:cons_f2_0(b)) ->_IH
*3_0

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(58)
Complex Obligation (BEST)

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

(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
encArg(a) -> a
encArg(g(x_1)) -> g(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_a -> a
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
a :: a:g:cons_f
g :: a:g:cons_f -> a:g:cons_f
encArg :: a:g:cons_f -> a:g:cons_f
cons_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_a :: a:g:cons_f
encode_g :: a:g:cons_f -> a:g:cons_f
hole_a:g:cons_f1_0 :: a:g:cons_f
gen_a:g:cons_f2_0 :: Nat -> a:g:cons_f


Generator Equations:
gen_a:g:cons_f2_0(0) <=> a
gen_a:g:cons_f2_0(+(x, 1)) <=> g(gen_a:g:cons_f2_0(x))


The following defined symbols remain to be analysed:
f, encArg

They will be analysed ascendingly in the following order:
f < encArg

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(60) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(61)
BOUNDS(n^1, INF)

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(62)
Obligation:
Innermost TRS:
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
encArg(a) -> a
encArg(g(x_1)) -> g(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_a -> a
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
a :: a:g:cons_f
g :: a:g:cons_f -> a:g:cons_f
encArg :: a:g:cons_f -> a:g:cons_f
cons_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_f :: a:g:cons_f -> a:g:cons_f -> a:g:cons_f
encode_a :: a:g:cons_f
encode_g :: a:g:cons_f -> a:g:cons_f
hole_a:g:cons_f1_0 :: a:g:cons_f
gen_a:g:cons_f2_0 :: Nat -> a:g:cons_f


Lemmas:
f(gen_a:g:cons_f2_0(+(1, n4_0)), gen_a:g:cons_f2_0(b)) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_a:g:cons_f2_0(0) <=> a
gen_a:g:cons_f2_0(+(x, 1)) <=> g(gen_a:g:cons_f2_0(x))


The following defined symbols remain to be analysed:
encArg
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(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:g:cons_f2_0(n426_0)) -> gen_a:g:cons_f2_0(n426_0), rt in Omega(0)

Induction Base:
encArg(gen_a:g:cons_f2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_a:g:cons_f2_0(+(n426_0, 1))) ->_R^Omega(0)
g(encArg(gen_a:g:cons_f2_0(n426_0))) ->_IH
g(gen_a:g:cons_f2_0(c427_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(64)
BOUNDS(1, INF)