/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, EXP).

(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) 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), 243 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 109 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 200 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 211 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, EXP)
(41) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(42) TRS for Loop Detection
    (43) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (44) BEST
        (45) proven lower bound
            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
            (47) BOUNDS(n^1, INF)
        (48) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   f(X) -> cons(X, n__f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(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, EXP).


The TRS R consists of the following rules:

   f(X) -> cons(X, n__f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   f(X) -> n__f(X) [1]
   activate(n__f(X)) -> f(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:

   f(X) -> cons(X, n__f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   f(X) -> n__f(X) [1]
   activate(n__f(X)) -> f(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
f :: 0:s -> n__f:cons
cons :: 0:s -> n__f:cons -> n__f:cons
n__f :: 0:s -> n__f:cons
g :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sel :: 0:s -> n__f:cons -> 0:s
activate :: n__f:cons -> n__f: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
   f_1
   g_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:

   f(X) -> cons(X, n__f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   f(X) -> n__f(X) [1]
   activate(n__f(X)) -> f(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
f :: 0:s -> n__f:cons
cons :: 0:s -> n__f:cons -> n__f:cons
n__f :: 0:s -> n__f:cons
g :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sel :: 0:s -> n__f:cons -> 0:s
activate :: n__f:cons -> n__f:cons
const :: n__f: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:

   f(X) -> cons(X, n__f(g(X))) [1]
   g(0) -> s(0) [1]
   g(s(X)) -> s(s(g(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, n__f(X'))) -> sel(X, f(X')) [2]
   sel(s(X), cons(Y, Z)) -> sel(X, Z) [2]
   f(X) -> n__f(X) [1]
   activate(n__f(X)) -> f(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
f :: 0:s -> n__f:cons
cons :: 0:s -> n__f:cons -> n__f:cons
n__f :: 0:s -> n__f:cons
g :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sel :: 0:s -> n__f:cons -> 0:s
activate :: n__f:cons -> n__f:cons
const :: n__f: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 }-> f(X) :|: z = 1 + X, X >= 0
   f(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   f(z) -{ 1 }-> 1 + X + (1 + g(X)) :|: X >= 0, z = X
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + g(X)) :|: z = 1 + X, X >= 0
   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, f(X')) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 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 }-> f(z - 1) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 1 }-> 1 + z + (1 + g(z)) :|: z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 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') -{ 2 }-> sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0


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

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

   { g }
   { f }
   { activate }
   { sel }

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

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

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

Function symbols to be analyzed: {g}, {f}, {activate}, {sel}

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

(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 }-> f(z - 1) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 1 }-> 1 + z + (1 + g(z)) :|: z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 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') -{ 2 }-> sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

Function symbols to be analyzed: {g}, {f}, {activate}, {sel}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: g
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 }-> f(z - 1) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 1 }-> 1 + z + (1 + g(z)) :|: z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + g(z - 1)) :|: z - 1 >= 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') -{ 2 }-> sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> f(z - 1) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 2 }-> sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> f(z - 1) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 2 }-> sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

Function symbols to be analyzed: {activate}, {sel}
Previous analysis results are:
  g: runtime: O(n^1) [1 + z], size: O(n^1) [1 + 2*z]
  f: runtime: O(n^1) [2 + z], size: O(n^1) [3 + 3*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) -{ 2 + z }-> s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 4 + X' }-> sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

(29) 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: 3*z

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

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

   activate(z) -{ 2 + z }-> s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 4 + X' }-> sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 2 + z }-> s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 4 + X' }-> sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(34)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 2 + z }-> s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 4 + X' }-> sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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(35) 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: ?

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

   activate(z) -{ 2 + z }-> s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 4 + X' }-> sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   activate(z) -{ 2 + z }-> s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
   activate(z) -{ 1 }-> z :|: z >= 0
   f(z) -{ 1 }-> 1 + z :|: z >= 0
   f(z) -{ 2 + z }-> 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
   g(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 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') -{ 4 + X' }-> sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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(39) FinalProof (FINISHED)
Computed overall runtime complexity
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(40)
BOUNDS(1, EXP)

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(41) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(42)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(X) -> cons(X, n__f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X

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

The rewrite sequence

g(s(X)) ->^+ s(s(g(X)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].

The pumping substitution is [X / s(X)].

The result substitution is [ ].




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(44)
Complex Obligation (BEST)

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(45)
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, EXP).


The TRS R consists of the following rules:

   f(X) -> cons(X, n__f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
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(46) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(47)
BOUNDS(n^1, INF)

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(48)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(X) -> cons(X, n__f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST