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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, 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), 129 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), 99 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 7 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 207 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 57 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
    (34) CpxRNTS
    (35) FinalProof [FINISHED, 0 ms]
    (36) BOUNDS(1, n^1)
(37) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(38) TRS for Loop Detection
    (39) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (40) BEST
        (41) proven lower bound
            (42) LowerBoundPropagationProof [FINISHED, 0 ms]
            (43) BOUNDS(n^1, INF)
        (44) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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:

   from(X) -> cons(X, n__from(s(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [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:

   from(X) -> cons(X, n__from(s(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:cons
cons :: s:0 -> n__from:cons -> n__from:cons
n__from :: s:0 -> n__from:cons
s :: s:0 -> s:0
sel :: s:0 -> n__from:cons -> s:0
0 :: s:0
activate :: n__from:cons -> n__from:cons

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:

   sel_2

(c) The following functions are completely defined:

   activate_1
   from_1

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:

   from(X) -> cons(X, n__from(s(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:cons
cons :: s:0 -> n__from:cons -> n__from:cons
n__from :: s:0 -> n__from:cons
s :: s:0 -> s:0
sel :: s:0 -> n__from:cons -> s:0
0 :: s:0
activate :: n__from:cons -> n__from:cons
const :: n__from:cons

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:

   from(X) -> cons(X, n__from(s(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, n__from(X'))) -> sel(X, from(X')) [2]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [2]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:cons
cons :: s:0 -> n__from:cons -> n__from:cons
n__from :: s:0 -> n__from:cons
s :: s:0 -> s:0
sel :: s:0 -> n__from:cons -> s:0
0 :: s:0
activate :: n__from:cons -> n__from:cons
const :: n__from:cons

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
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 2 }-> sel(X, from(X')) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0


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

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

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

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(X, 1 + X'') :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(X, 1 + X'' + (1 + (1 + X''))) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''


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

(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
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''


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

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

   { activate }
   { sel }
   { from }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {activate}, {sel}, {from}

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

(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
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {activate}, {sel}, {from}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {sel}, {from}
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^1) [1 + 2*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
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {sel}, {from}
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^1) [1 + 2*z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: sel
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {sel}, {from}
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^1) [1 + 2*z]
  sel: runtime: ?, size: EXP

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sel
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 8*z

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 2 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ 3 }-> sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {from}
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^1) [1 + 2*z]
  sel: runtime: O(n^1) [1 + 8*z], size: EXP

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

(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
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ -5 + 8*z }-> s :|: s >= 0, s <= inf, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ -4 + 8*z }-> s' :|: s' >= 0, s' <= inf', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ -4 + 8*z }-> s'' :|: s'' >= 0, s'' <= inf'', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {from}
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^1) [1 + 2*z]
  sel: runtime: O(n^1) [1 + 8*z], size: EXP

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ -5 + 8*z }-> s :|: s >= 0, s <= inf, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ -4 + 8*z }-> s' :|: s' >= 0, s' <= inf', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ -4 + 8*z }-> s'' :|: s'' >= 0, s'' <= inf'', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {from}
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^1) [1 + 2*z]
  sel: runtime: O(n^1) [1 + 8*z], size: EXP
  from: runtime: ?, size: O(n^1) [3 + 2*z]

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(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   activate(z) -{ 2 }-> 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ -5 + 8*z }-> s :|: s >= 0, s <= inf, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
   sel(z, z') -{ -4 + 8*z }-> s' :|: s' >= 0, s' <= inf', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
   sel(z, z') -{ -4 + 8*z }-> s'' :|: s'' >= 0, s'' <= inf'', Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: 
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^1) [1 + 2*z]
  sel: runtime: O(n^1) [1 + 8*z], size: EXP
  from: runtime: O(1) [1], size: O(n^1) [3 + 2*z]

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

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(37) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(38)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
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(39) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

sel(s(X), cons(Y, Z)) ->^+ sel(X, Z)

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

The pumping substitution is [X / s(X), Z / cons(Y, Z)].

The result substitution is [ ].




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(40)
Complex Obligation (BEST)

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(41)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
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(42) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(43)
BOUNDS(n^1, INF)

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(44)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST