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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) 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), 36 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
    (16) CpxRNTS
    (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 253 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 68 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 252 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 179 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 191 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 259 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (50) CpxRNTS
    (51) FinalProof [FINISHED, 0 ms]
    (52) BOUNDS(1, n^2)
(53) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(54) TRS for Loop Detection
    (55) DecreasingLoopProof [LOWER BOUND(ID), 3 ms]
    (56) BEST
        (57) proven lower bound
            (58) LowerBoundPropagationProof [FINISHED, 0 ms]
            (59) BOUNDS(n^1, INF)
        (60) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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

(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(b(x_1)) -> b(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

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


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(5) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x)) [1]
   a(c(x)) -> x [1]
   encArg(b(x_1)) -> b(encArg(x_1)) [0]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]

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

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

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

   a(b(x)) -> b(a(x)) [1]
   a(c(x)) -> x [1]
   encArg(b(x_1)) -> b(encArg(x_1)) [0]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]

The TRS has the following type information:
a :: b:c:cons_a -> b:c:cons_a
b :: b:c:cons_a -> b:c:cons_a
c :: b:c:cons_a -> b:c:cons_a
encArg :: b:c:cons_a -> b:c:cons_a
cons_a :: b:c:cons_a -> b:c:cons_a
encode_a :: b:c:cons_a -> b:c:cons_a
encode_b :: b:c:cons_a -> b:c:cons_a
encode_c :: b:c:cons_a -> b:c:cons_a

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

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

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

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

(c) The following functions are completely defined:

   a_1
   encArg_1
   encode_a_1
   encode_b_1
   encode_c_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_b(v0) -> const [0]
   encode_c(v0) -> const [0]
   a(v0) -> const [0]

And the following fresh constants:    const

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

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

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

   a(b(x)) -> b(a(x)) [1]
   a(c(x)) -> x [1]
   encArg(b(x_1)) -> b(encArg(x_1)) [0]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(x_1)) -> a(encArg(x_1)) [0]
   encode_a(x_1) -> a(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_b(v0) -> const [0]
   encode_c(v0) -> const [0]
   a(v0) -> const [0]

The TRS has the following type information:
a :: b:c:cons_a:const -> b:c:cons_a:const
b :: b:c:cons_a:const -> b:c:cons_a:const
c :: b:c:cons_a:const -> b:c:cons_a:const
encArg :: b:c:cons_a:const -> b:c:cons_a:const
cons_a :: b:c:cons_a:const -> b:c:cons_a:const
encode_a :: b:c:cons_a:const -> b:c:cons_a:const
encode_b :: b:c:cons_a:const -> b:c:cons_a:const
encode_c :: b:c:cons_a:const -> b:c:cons_a:const
const :: b:c:cons_a:const

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   a(b(x)) -> b(a(x)) [1]
   a(c(x)) -> x [1]
   encArg(b(x_1)) -> b(encArg(x_1)) [0]
   encArg(c(x_1)) -> c(encArg(x_1)) [0]
   encArg(cons_a(b(x_1'))) -> a(b(encArg(x_1'))) [0]
   encArg(cons_a(c(x_1''))) -> a(c(encArg(x_1''))) [0]
   encArg(cons_a(cons_a(x_11))) -> a(a(encArg(x_11))) [0]
   encArg(cons_a(x_1)) -> a(const) [0]
   encode_a(b(x_12)) -> a(b(encArg(x_12))) [0]
   encode_a(c(x_13)) -> a(c(encArg(x_13))) [0]
   encode_a(cons_a(x_14)) -> a(a(encArg(x_14))) [0]
   encode_a(x_1) -> a(const) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   encode_c(x_1) -> c(encArg(x_1)) [0]
   encArg(v0) -> const [0]
   encode_a(v0) -> const [0]
   encode_b(v0) -> const [0]
   encode_c(v0) -> const [0]
   a(v0) -> const [0]

The TRS has the following type information:
a :: b:c:cons_a:const -> b:c:cons_a:const
b :: b:c:cons_a:const -> b:c:cons_a:const
c :: b:c:cons_a:const -> b:c:cons_a:const
encArg :: b:c:cons_a:const -> b:c:cons_a:const
cons_a :: b:c:cons_a:const -> b:c:cons_a:const
encode_a :: b:c:cons_a:const -> b:c:cons_a:const
encode_b :: b:c:cons_a:const -> b:c:cons_a:const
encode_c :: b:c:cons_a:const -> b:c:cons_a:const
const :: b:c:cons_a:const

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

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

   const => 0

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

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

   a(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   a(z) -{ 1 }-> 1 + a(x) :|: x >= 0, z = 1 + x
   encArg(z) -{ 0 }-> a(a(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11)
   encArg(z) -{ 0 }-> a(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(x_1'')) :|: z = 1 + (1 + x_1''), x_1'' >= 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(z) -{ 0 }-> a(a(encArg(x_14))) :|: x_14 >= 0, z = 1 + x_14
   encode_a(z) -{ 0 }-> a(0) :|: x_1 >= 0, z = x_1
   encode_a(z) -{ 0 }-> a(1 + encArg(x_12)) :|: z = 1 + x_12, x_12 >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(x_13)) :|: z = 1 + x_13, x_13 >= 0
   encode_a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_c(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_c(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1


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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 }-> 1 + a(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: z >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0


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

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

   { a }
   { encArg }
   { encode_c }
   { encode_b }
   { encode_a }

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 }-> 1 + a(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: z >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0

Function symbols to be analyzed: {a}, {encArg}, {encode_c}, {encode_b}, {encode_a}

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 }-> 1 + a(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: z >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0

Function symbols to be analyzed: {a}, {encArg}, {encode_c}, {encode_b}, {encode_a}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 }-> 1 + a(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: z >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0

Function symbols to be analyzed: {a}, {encArg}, {encode_c}, {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: ?, size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 }-> 1 + a(z - 1) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(0) :|: z >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_c}, {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0

Function symbols to be analyzed: {encArg}, {encode_c}, {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 0 }-> a(a(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> a(1 + encArg(z - 2)) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 0 }-> a(a(encArg(z - 1))) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> a(1 + encArg(z - 1)) :|: z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0

Function symbols to be analyzed: {encode_c}, {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_c}, {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_c}, {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: ?, size: O(n^1) [1 + z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_b}, {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]
  encode_b: runtime: ?, size: O(n^1) [1 + z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]
  encode_b: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]
  encode_b: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: {encode_a}
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]
  encode_b: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]
  encode_a: runtime: ?, size: O(n^1) [z]

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(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: encode_a
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 13 + 6*z^2

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

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   a(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   encArg(z) -{ 1 }-> s' :|: s' >= 0, s' <= 0, z - 1 >= 0
   encArg(z) -{ 15 + s2 + -12*z + 3*z^2 }-> s3 :|: s2 >= 0, s2 <= z - 2, s3 >= 0, s3 <= 1 + s2, z - 2 >= 0
   encArg(z) -{ 15 + s4 + s5 + -12*z + 3*z^2 }-> s6 :|: s4 >= 0, s4 <= z - 2, s5 >= 0, s5 <= s4, s6 >= 0, s6 <= s5, z - 2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 4 + -6*z + 3*z^2 }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0
   encode_a(z) -{ 1 }-> s'' :|: s'' >= 0, s'' <= 0, z >= 0
   encode_a(z) -{ 6 + s10 + s9 + -6*z + 3*z^2 }-> s11 :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= s9, s11 >= 0, s11 <= s10, z - 1 >= 0
   encode_a(z) -{ 6 + s7 + -6*z + 3*z^2 }-> s8 :|: s7 >= 0, s7 <= z - 1, s8 >= 0, s8 <= 1 + s7, z - 1 >= 0
   encode_a(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 0 }-> 0 :|: z >= 0
   encode_b(z) -{ 1 + 3*z^2 }-> 1 + s12 :|: s12 >= 0, s12 <= z, z >= 0
   encode_c(z) -{ 0 }-> 0 :|: z >= 0
   encode_c(z) -{ 1 + 3*z^2 }-> 1 + s13 :|: s13 >= 0, s13 <= z, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  a: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  encArg: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [z]
  encode_c: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]
  encode_b: runtime: O(n^2) [1 + 3*z^2], size: O(n^1) [1 + z]
  encode_a: runtime: O(n^2) [13 + 6*z^2], size: O(n^1) [z]

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(51) FinalProof (FINISHED)
Computed overall runtime complexity
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(52)
BOUNDS(1, n^2)

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(53) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(54)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

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

The rewrite sequence

a(b(x)) ->^+ b(a(x))

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

The pumping substitution is [x / b(x)].

The result substitution is [ ].




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(56)
Complex Obligation (BEST)

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(57)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: FULL
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(58) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(59)
BOUNDS(n^1, INF)

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(60)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: FULL