WORST_CASE(?, 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 148 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedTrs
(9) CompletionProof [UPPER BOUND(ID), 0 ms]
(10) CpxTypedWeightedCompleteTrs
(11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(14) CpxRNTS
(15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRNTS
(19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 259 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 201 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 176 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 162 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 138 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(42) CpxRNTS
(43) FinalProof [FINISHED, 0 ms]
(44) BOUNDS(1, n^2)


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

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


The TRS R consists of the following rules:

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

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

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))


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

(2)
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:

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

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   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(1, n^2).


The TRS R consists of the following rules:

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

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   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(g(x), y, y) -> g(f(x, x, y)) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [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(g(x), y, y) -> g(f(x, x, y)) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]

The TRS has the following type information:
f :: g:cons_f -> g:cons_f -> g:cons_f -> g:cons_f
g :: g:cons_f -> g:cons_f
encArg :: g:cons_f -> g:cons_f
cons_f :: g:cons_f -> g:cons_f -> g:cons_f -> g:cons_f
encode_f :: g:cons_f -> g:cons_f -> g:cons_f -> g:cons_f
encode_g :: g:cons_f -> 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_3
   encArg_1
   encode_f_3
   encode_g_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_f(v0, v1, v2) -> const [0]
   encode_g(v0) -> const [0]
   f(v0, v1, v2) -> const [0]

And the following fresh constants:    const

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

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

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

   f(g(x), y, y) -> g(f(x, x, y)) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1, v2) -> const [0]
   encode_g(v0) -> const [0]
   f(v0, v1, v2) -> const [0]

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

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

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

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

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

   f(g(x), y, y) -> g(f(x, x, y)) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1, v2) -> const [0]
   encode_g(v0) -> const [0]
   f(v0, v1, v2) -> const [0]

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

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

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

   const => 0

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_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', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   f(z, z', z'') -{ 1 }-> 1 + f(x, x, y) :|: z'' = y, x >= 0, y >= 0, z = 1 + x, z' = y


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

(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(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 1 }-> 1 + f(z - 1, z - 1, z'') :|: z - 1 >= 0, z'' >= 0, z' = z''


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

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

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

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 1 }-> 1 + f(z - 1, z - 1, z'') :|: z - 1 >= 0, z'' >= 0, z' = z''

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

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 1 }-> 1 + f(z - 1, z - 1, z'') :|: z - 1 >= 0, z'' >= 0, z' = z''

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 1 }-> 1 + f(z - 1, z - 1, z'') :|: z - 1 >= 0, z'' >= 0, z' = z''

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
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:

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ 1 }-> 1 + f(z - 1, z - 1, z'') :|: z - 1 >= 0, z'' >= 0, z' = z''

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

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

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0
   encode_f(z, z', z'') -{ 0 }-> f(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

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

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

   encArg(z) -{ s'' + x_1 + x_1^2 + x_2 + x_2^2 + x_3 + x_3^2 }-> s3 :|: s'' >= 0, s'' <= x_1, s1 >= 0, s1 <= x_2, s2 >= 0, s2 <= x_3, s3 >= 0, s3 <= s'', x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -1*z + z^2 }-> 1 + s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0
   encode_f(z, z', z'') -{ s4 + z + z^2 + z' + z'^2 + z'' + z''^2 }-> s7 :|: s4 >= 0, s4 <= z, s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= s4, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ z + z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z, z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   encArg(z) -{ s'' + x_1 + x_1^2 + x_2 + x_2^2 + x_3 + x_3^2 }-> s3 :|: s'' >= 0, s'' <= x_1, s1 >= 0, s1 <= x_2, s2 >= 0, s2 <= x_3, s3 >= 0, s3 <= s'', x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -1*z + z^2 }-> 1 + s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0
   encode_f(z, z', z'') -{ s4 + z + z^2 + z' + z'^2 + z'' + z''^2 }-> s7 :|: s4 >= 0, s4 <= z, s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= s4, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ z + z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z, z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

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

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

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

   encArg(z) -{ s'' + x_1 + x_1^2 + x_2 + x_2^2 + x_3 + x_3^2 }-> s3 :|: s'' >= 0, s'' <= x_1, s1 >= 0, s1 <= x_2, s2 >= 0, s2 <= x_3, s3 >= 0, s3 <= s'', x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -1*z + z^2 }-> 1 + s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0
   encode_f(z, z', z'') -{ s4 + z + z^2 + z' + z'^2 + z'' + z''^2 }-> s7 :|: s4 >= 0, s4 <= z, s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= s4, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ z + z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z, z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

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

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

   encArg(z) -{ s'' + x_1 + x_1^2 + x_2 + x_2^2 + x_3 + x_3^2 }-> s3 :|: s'' >= 0, s'' <= x_1, s1 >= 0, s1 <= x_2, s2 >= 0, s2 <= x_3, s3 >= 0, s3 <= s'', x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -1*z + z^2 }-> 1 + s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0
   encode_f(z, z', z'') -{ s4 + z + z^2 + z' + z'^2 + z'' + z''^2 }-> s7 :|: s4 >= 0, s4 <= z, s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= s4, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ z + z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z, z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

(39) 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

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

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

   encArg(z) -{ s'' + x_1 + x_1^2 + x_2 + x_2^2 + x_3 + x_3^2 }-> s3 :|: s'' >= 0, s'' <= x_1, s1 >= 0, s1 <= x_2, s2 >= 0, s2 <= x_3, s3 >= 0, s3 <= s'', x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -1*z + z^2 }-> 1 + s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0
   encode_f(z, z', z'') -{ s4 + z + z^2 + z' + z'^2 + z'' + z''^2 }-> s7 :|: s4 >= 0, s4 <= z, s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= s4, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ z + z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z, z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

(41) 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: z + z^2

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

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

   encArg(z) -{ s'' + x_1 + x_1^2 + x_2 + x_2^2 + x_3 + x_3^2 }-> s3 :|: s'' >= 0, s'' <= x_1, s1 >= 0, s1 <= x_2, s2 >= 0, s2 <= x_3, s3 >= 0, s3 <= s'', x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encArg(z) -{ -1*z + z^2 }-> 1 + s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0
   encode_f(z, z', z'') -{ s4 + z + z^2 + z' + z'^2 + z'' + z''^2 }-> s7 :|: s4 >= 0, s4 <= z, s5 >= 0, s5 <= z', s6 >= 0, s6 <= z'', s7 >= 0, s7 <= s4, z >= 0, z'' >= 0, z' >= 0
   encode_f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   encode_g(z) -{ 0 }-> 0 :|: z >= 0
   encode_g(z) -{ z + z^2 }-> 1 + s8 :|: s8 >= 0, s8 <= z, z >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   f(z, z', z'') -{ z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0, z'' >= 0, z' = z''

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

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

(44)
BOUNDS(1, n^2)