WORST_CASE(?,O(n^1))
proof of input_Aej0q87UuZ.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxTypedWeightedTrs
(5) CompletionProof [UPPER BOUND(ID), 0 ms]
(6) CpxTypedWeightedCompleteTrs
(7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(10) CpxRNTS
(11) InliningProof [UPPER BOUND(ID), 38 ms]
(12) CpxRNTS
(13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 191 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 363 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 539 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
(34) CpxRNTS
(35) FinalProof [FINISHED, 0 ms]
(36) BOUNDS(1, n^1)


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   prime(0) -> false [1]
   prime(s(0)) -> false [1]
   prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1]
   prime1(x, 0) -> false [1]
   prime1(x, s(0)) -> true [1]
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
   divp(x, y) -> =(rem(x, y), 0) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   prime(0) -> false [1]
   prime(s(0)) -> false [1]
   prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1]
   prime1(x, 0) -> false [1]
   prime1(x, s(0)) -> true [1]
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
   divp(x, y) -> =(rem(x, y), 0) [1]

The TRS has the following type information:
prime :: 0:s -> false:true:and
0 :: 0:s
false :: false:true:and
s :: 0:s -> 0:s
prime1 :: 0:s -> 0:s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: = -> not
divp :: 0:s -> 0:s -> =
= :: rem -> 0:s -> =
rem :: 0:s -> 0:s -> rem

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

(5) 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:

   prime_1
   prime1_2
   divp_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2

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

(6)
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:

   prime(0) -> false [1]
   prime(s(0)) -> false [1]
   prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1]
   prime1(x, 0) -> false [1]
   prime1(x, s(0)) -> true [1]
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
   divp(x, y) -> =(rem(x, y), 0) [1]

The TRS has the following type information:
prime :: 0:s -> false:true:and
0 :: 0:s
false :: false:true:and
s :: 0:s -> 0:s
prime1 :: 0:s -> 0:s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: = -> not
divp :: 0:s -> 0:s -> =
= :: rem -> 0:s -> =
rem :: 0:s -> 0:s -> rem
const :: not
const1 :: =
const2 :: rem

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

(8)
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:

   prime(0) -> false [1]
   prime(s(0)) -> false [1]
   prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1]
   prime1(x, 0) -> false [1]
   prime1(x, s(0)) -> true [1]
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
   divp(x, y) -> =(rem(x, y), 0) [1]

The TRS has the following type information:
prime :: 0:s -> false:true:and
0 :: 0:s
false :: false:true:and
s :: 0:s -> 0:s
prime1 :: 0:s -> 0:s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: = -> not
divp :: 0:s -> 0:s -> =
= :: rem -> 0:s -> =
rem :: 0:s -> 0:s -> rem
const :: not
const1 :: =
const2 :: rem

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

   0 => 0
   false => 0
   true => 1
   const => 0
   const1 => 0
   const2 => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   divp(z, z') -{ 1 }-> 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y
   prime(z) -{ 1 }-> prime1(1 + (1 + x), 1 + x) :|: x >= 0, z = 1 + (1 + x)
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1 + 0, z = x
   prime1(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   prime1(z, z') -{ 1 }-> 1 + (1 + divp(1 + (1 + y), x)) + prime1(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x


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(11) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   divp(z, z') -{ 1 }-> 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   divp(z, z') -{ 1 }-> 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y
   prime(z) -{ 1 }-> prime1(1 + (1 + x), 1 + x) :|: x >= 0, z = 1 + (1 + x)
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1 + 0, z = x
   prime1(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + y') + 0)) + prime1(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x, x' >= 0, y' >= 0, 1 + (1 + y) = x', x = y'


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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'


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

   { divp }
   { prime1 }
   { prime }

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {divp}, {prime1}, {prime}

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {divp}, {prime1}, {prime}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {divp}, {prime1}, {prime}
Previous analysis results are:
  divp: runtime: ?, size: O(n^1) [2 + z + z']

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime1}, {prime}
Previous analysis results are:
  divp: runtime: O(1) [1], size: O(n^1) [2 + z + z']

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime1}, {prime}
Previous analysis results are:
  divp: runtime: O(1) [1], size: O(n^1) [2 + z + z']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: prime1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + z*z' + 4*z' + z'^2

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime1}, {prime}
Previous analysis results are:
  divp: runtime: O(1) [1], size: O(n^1) [2 + z + z']
  prime1: runtime: ?, size: O(n^2) [1 + z*z' + 4*z' + z'^2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: prime1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 2*z'

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 }-> prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime}
Previous analysis results are:
  divp: runtime: O(1) [1], size: O(n^1) [2 + z + z']
  prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2]

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (z - 2)) * (1 + (1 + (z - 2))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 + 2*z' }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * (1 + (z' - 2)) + (1 + (z' - 2)) * z + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime}
Previous analysis results are:
  divp: runtime: O(1) [1], size: O(n^1) [2 + z + z']
  prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2]

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(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (z - 2)) * (1 + (1 + (z - 2))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 + 2*z' }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * (1 + (z' - 2)) + (1 + (z' - 2)) * z + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime}
Previous analysis results are:
  divp: runtime: O(1) [1], size: O(n^1) [2 + z + z']
  prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2]
  prime: runtime: ?, size: O(n^2) [2 + z + 2*z^2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
   prime(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (z - 2)) * (1 + (1 + (z - 2))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 1 + 0
   prime1(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   prime1(z, z') -{ 2 + 2*z' }-> 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * (1 + (z' - 2)) + (1 + (z' - 2)) * z + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: 
Previous analysis results are:
  divp: runtime: O(1) [1], size: O(n^1) [2 + z + z']
  prime1: runtime: O(n^1) [2 + 2*z'], size: O(n^2) [1 + z*z' + 4*z' + z'^2]
  prime: runtime: O(n^1) [1 + 2*z], size: O(n^2) [2 + z + 2*z^2]

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(35) FinalProof (FINISHED)
Computed overall runtime complexity
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(36)
BOUNDS(1, n^1)