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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) InliningProof [UPPER BOUND(ID), 81 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 199 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 35 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 486 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 46 ms]
    (30) CpxRNTS
    (31) FinalProof [FINISHED, 0 ms]
    (32) BOUNDS(1, n^1)
(33) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(34) TRS for Loop Detection
    (35) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (36) BEST
        (37) proven lower bound
            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
            (39) BOUNDS(n^1, INF)
        (40) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(4)
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:

   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   from(X) -> cons(X, n__from(s(X))) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   from(X) -> n__from(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   from(X) -> cons(X, n__from(s(X))) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   from(X) -> n__from(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

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

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

(7) 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:

   first_2
   from_1
   activate_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   from(X) -> cons(X, n__from(s(X))) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   from(X) -> n__from(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

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

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

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

(10)
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:

   first(0, X) -> nil [1]
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z))) [1]
   from(X) -> cons(X, n__from(s(X))) [1]
   first(X1, X2) -> n__first(X1, X2) [1]
   from(X) -> n__from(X) [1]
   activate(n__first(X1, X2)) -> first(X1, X2) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

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

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

(11) 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

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

(12)
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
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X


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

(13) 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

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X


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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0


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

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

   { from }
   { first, activate }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

(21) 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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

(23) 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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: first
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 2*z'

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 9 + 2*z

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   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'
   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
   from(z) -{ 1 }-> 1 + z :|: z >= 0
   from(z) -{ 1 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [3 + 2*z]
  first: runtime: O(n^1) [5 + 2*z'], size: O(n^1) [1 + z + 2*z']
  activate: runtime: O(n^1) [9 + 2*z], size: O(n^1) [1 + 2*z]

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

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

(32)
BOUNDS(1, n^1)

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

(33) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(34)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(35) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

activate(n__first(s(X1_0), cons(Y2_0, Z3_0))) ->^+ cons(Y2_0, n__first(X1_0, activate(Z3_0)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].

The pumping substitution is [Z3_0 / n__first(s(X1_0), cons(Y2_0, Z3_0))].

The result substitution is [ ].




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

(36)
Complex Obligation (BEST)

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

(37)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(38) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(39)
BOUNDS(n^1, INF)

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(40)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL