WORST_CASE(?,O(n^3))
proof of input_hWE3S4JUFE.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^3).

(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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxRNTS
(13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) IntTrsBoundProof [UPPER BOUND(ID), 244 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 284 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 368 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
(32) CpxRNTS
(33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 530 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), 1979 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 29 ms]
(44) CpxRNTS
(45) FinalProof [FINISHED, 0 ms]
(46) BOUNDS(1, n^3)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   terms(X) -> n__terms(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   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^3).


The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   terms(X) -> n__terms(X) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   activate(n__terms(X)) -> terms(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [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:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   terms(X) -> n__terms(X) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   activate(n__terms(X)) -> terms(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
terms :: s:0 -> n__terms:cons:nil:n__first
cons :: recip -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
recip :: s:0 -> recip
sqr :: s:0 -> s:0
n__terms :: s:0 -> n__terms:cons:nil:n__first
s :: s:0 -> s:0
0 :: s:0
add :: s:0 -> s:0 -> s:0
dbl :: s:0 -> s:0
first :: s:0 -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0 -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first

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:

   terms_1
   first_2
   activate_1

(c) The following functions are completely defined:

   sqr_1
   dbl_1
   add_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:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(X)) -> s(add(sqr(X), dbl(X))) [1]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   terms(X) -> n__terms(X) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   activate(n__terms(X)) -> terms(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
terms :: s:0 -> n__terms:cons:nil:n__first
cons :: recip -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
recip :: s:0 -> recip
sqr :: s:0 -> s:0
n__terms :: s:0 -> n__terms:cons:nil:n__first
s :: s:0 -> s:0
0 :: s:0
add :: s:0 -> s:0 -> s:0
dbl :: s:0 -> s:0
first :: s:0 -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0 -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
const :: recip

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:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N))) [1]
   sqr(0) -> 0 [1]
   sqr(s(0)) -> s(add(0, 0)) [3]
   sqr(s(s(X'))) -> s(add(s(add(sqr(X'), dbl(X'))), s(s(dbl(X'))))) [3]
   dbl(0) -> 0 [1]
   dbl(s(X)) -> s(s(dbl(X))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   terms(X) -> n__terms(X) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   activate(n__terms(X)) -> terms(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
terms :: s:0 -> n__terms:cons:nil:n__first
cons :: recip -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
recip :: s:0 -> recip
sqr :: s:0 -> s:0
n__terms :: s:0 -> n__terms:cons:nil:n__first
s :: s:0 -> s:0
0 :: s:0
add :: s:0 -> s:0 -> s:0
dbl :: s:0 -> s:0
first :: s:0 -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0 -> n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first -> n__terms:cons:nil:n__first
const :: recip

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:

   0 => 0
   nil => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> terms(X) :|: z = 1 + X, X >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0
   first(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   first(z, z') -{ 1 }-> 1 + Y + (1 + X + activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(X'), dbl(X')), 1 + (1 + dbl(X'))) :|: X' >= 0, z = 1 + (1 + X')
   terms(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   terms(z) -{ 1 }-> 1 + (1 + sqr(N)) + (1 + (1 + N)) :|: z = N, N >= 0


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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0


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

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

   { dbl }
   { add }
   { sqr }
   { terms }
   { first, activate }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}, {first,activate}

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}, {first,activate}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {dbl}, {add}, {sqr}, {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: ?, size: O(n^1) [2*z]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 3 }-> 1 + add(1 + add(sqr(z - 2), dbl(z - 2)), 1 + (1 + dbl(z - 2))) :|: z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {add}, {sqr}, {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: ?, size: O(n^1) [z + z']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: add
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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 3 }-> 1 + add(0, 0) :|: z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {sqr}, {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + 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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {sqr}, {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {sqr}, {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: ?, size: O(n^2) [1 + 4*z + 4*z^2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 5 + 22*z + 8*z^3

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ 1 + 2*z }-> 1 + add(1 + add(sqr(z - 2), s), 1 + (1 + s')) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 1 }-> 1 + (1 + sqr(z)) + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]

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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) + (1 + (1 + z)) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: terms
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 6*z + 4*z^2

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) + (1 + (1 + z)) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {terms}, {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: ?, size: O(n^2) [6 + 6*z + 4*z^2]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: terms
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 7 + 22*z + 8*z^3

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> terms(z - 1) :|: z - 1 >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) + (1 + (1 + z)) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: O(n^3) [7 + 22*z + 8*z^3], size: O(n^2) [6 + 6*z + 4*z^2]

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

(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:

   activate(z) -{ -22 + 46*z + -24*z^2 + 8*z^3 }-> s7 :|: s7 >= 0, s7 <= 6 * (z - 1) + 4 * ((z - 1) * (z - 1)) + 6, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) + (1 + (1 + z)) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: O(n^3) [7 + 22*z + 8*z^3], size: O(n^2) [6 + 6*z + 4*z^2]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: first
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   activate(z) -{ -22 + 46*z + -24*z^2 + 8*z^3 }-> s7 :|: s7 >= 0, s7 <= 6 * (z - 1) + 4 * ((z - 1) * (z - 1)) + 6, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) + (1 + (1 + z)) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: {first,activate}
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: O(n^3) [7 + 22*z + 8*z^3], size: O(n^2) [6 + 6*z + 4*z^2]
  first: runtime: ?, size: INF
  activate: runtime: ?, size: INF

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: first
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 3 + 48*z' + 8*z'^3

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 5 + 94*z + 16*z^3

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

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

   activate(z) -{ -22 + 46*z + -24*z^2 + 8*z^3 }-> s7 :|: s7 >= 0, s7 <= 6 * (z - 1) + 4 * ((z - 1) * (z - 1)) + 6, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z' >= 0, z - 1 >= 0
   dbl(z) -{ 1 }-> 0 :|: z = 0
   dbl(z) -{ 1 + z }-> 1 + (1 + s'') :|: s'' >= 0, s'' <= 2 * (z - 1), z - 1 >= 0
   first(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   first(z, z') -{ 1 }-> 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   first(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   sqr(z) -{ 1 }-> 0 :|: z = 0
   sqr(z) -{ 4 }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + 0, z = 1 + 0
   sqr(z) -{ -99 + s4 + s5 + 120*z + -48*z^2 + 8*z^3 }-> 1 + s6 :|: s4 >= 0, s4 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s5 >= 0, s5 <= s4 + s, s6 >= 0, s6 <= 1 + s5 + (1 + (1 + s')), s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
   terms(z) -{ 1 }-> 1 + z :|: z >= 0
   terms(z) -{ 6 + 22*z + 8*z^3 }-> 1 + (1 + s3) + (1 + (1 + z)) :|: s3 >= 0, s3 <= 4 * (z * z) + 4 * z + 1, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  dbl: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  sqr: runtime: O(n^3) [5 + 22*z + 8*z^3], size: O(n^2) [1 + 4*z + 4*z^2]
  terms: runtime: O(n^3) [7 + 22*z + 8*z^3], size: O(n^2) [6 + 6*z + 4*z^2]
  first: runtime: O(n^3) [3 + 48*z' + 8*z'^3], size: INF
  activate: runtime: O(n^3) [5 + 94*z + 16*z^3], size: INF

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(45) FinalProof (FINISHED)
Computed overall runtime complexity
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(46)
BOUNDS(1, n^3)