WORST_CASE(?,O(n^2))
proof of input_jtuZRE3Jdb.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(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), 29 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), 148 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 192 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 302 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
(34) CpxRNTS
(35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 405 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 145 ms]
(40) CpxRNTS
(41) FinalProof [FINISHED, 0 ms]
(42) BOUNDS(1, n^2)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X)
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)
   activate(X) -> X

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

(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^2).


The TRS R consists of the following rules:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 0:s -> 0:s -> 0:s

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:

   and_2
   activate_1

(c) The following functions are completely defined:

   x_2
   plus_2

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(M)) -> plus(x(N, M), N) [1]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 0:s -> 0:s -> 0:s
const :: and:activate

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

(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:

   and(tt, X) -> activate(X) [1]
   plus(N, 0) -> N [1]
   plus(N, s(M)) -> s(plus(N, M)) [1]
   x(N, 0) -> 0 [1]
   x(N, s(0)) -> plus(0, N) [2]
   x(N, s(s(M'))) -> plus(plus(x(N, M'), N), N) [2]
   activate(X) -> X [1]

The TRS has the following type information:
and :: tt -> and:activate -> and:activate
tt :: tt
activate :: and:activate -> and:activate
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
x :: 0:s -> 0:s -> 0:s
const :: and:activate

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

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

   tt => 0
   0 => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   and(z, z') -{ 1 }-> activate(X) :|: z' = X, X >= 0, z = 0
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 2 }-> plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0
   x(z, z') -{ 2 }-> plus(0, N) :|: z = N, z' = 1 + 0, N >= 0
   x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0


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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   and(z, z') -{ 2 }-> X' :|: z' = X, X >= 0, z = 0, X' >= 0, X = X'
   plus(z, z') -{ 1 }-> N :|: z = N, z' = 0, N >= 0
   plus(z, z') -{ 1 }-> 1 + plus(N, M) :|: z' = 1 + M, z = N, M >= 0, N >= 0
   x(z, z') -{ 2 }-> plus(plus(x(N, M'), N), N) :|: M' >= 0, z = N, z' = 1 + (1 + M'), N >= 0
   x(z, z') -{ 2 }-> plus(0, N) :|: z = N, z' = 1 + 0, N >= 0
   x(z, z') -{ 1 }-> 0 :|: z = N, z' = 0, N >= 0


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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0


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

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

   { activate }
   { and }
   { plus }
   { x }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {and}, {plus}, {x}

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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {and}, {plus}, {x}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {activate}, {and}, {plus}, {x}
Previous analysis results are:
  activate: runtime: ?, size: O(n^1) [z]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: activate
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:

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {and}, {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]

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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {and}, {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {and}, {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: ?, size: O(n^1) [z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: and
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [z']

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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {plus}, {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 2 }-> plus(0, z) :|: z' = 1 + 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0
   x(z, z') -{ 3 + z }-> s' :|: s' >= 0, s' <= 0 + z, z' = 1 + 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0
   x(z, z') -{ 3 + z }-> s' :|: s' >= 0, s' <= 0 + z, z' = 1 + 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: {x}
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  x: runtime: ?, size: O(n^2) [z + 2*z*z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   and(z, z') -{ 2 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z' = 0, z >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z' - 1 >= 0, z >= 0
   x(z, z') -{ 3 + z }-> s' :|: s' >= 0, s' <= 0 + z, z' = 1 + 0, z >= 0
   x(z, z') -{ 2 }-> plus(plus(x(z, z' - 2), z), z) :|: z' - 2 >= 0, z >= 0
   x(z, z') -{ 1 }-> 0 :|: z' = 0, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  activate: runtime: O(1) [1], size: O(n^1) [z]
  and: runtime: O(1) [2], size: O(n^1) [z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  x: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z']

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

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

(42)
BOUNDS(1, n^2)