/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), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(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), 1 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), 4 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 336 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 361 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 601 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^2)
(35) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(36) TRS for Loop Detection
    (37) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (38) BEST
        (39) proven lower bound
            (40) LowerBoundPropagationProof [FINISHED, 0 ms]
            (41) BOUNDS(n^1, INF)
        (42) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   plus_x#1(0, x8) -> x8
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14))
   map#2(plus_x(x2), Nil) -> Nil
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
   main(x5, x12) -> map#2(plus_x(x12), x5)

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^2).


The TRS R consists of the following rules:

   plus_x#1(0, x8) -> x8 [1]
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14)) [1]
   map#2(plus_x(x2), Nil) -> Nil [1]
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1]
   main(x5, x12) -> map#2(plus_x(x12), x5) [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:

   plus_x#1(0, x8) -> x8 [1]
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14)) [1]
   map#2(plus_x(x2), Nil) -> Nil [1]
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1]
   main(x5, x12) -> map#2(plus_x(x12), x5) [1]

The TRS has the following type information:
plus_x#1 :: 0:S -> 0:S -> 0:S
0 :: 0:S
S :: 0:S -> 0:S
map#2 :: plus_x -> Nil:Cons -> Nil:Cons
plus_x :: 0:S -> plus_x
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> 0:S -> Nil: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:

   plus_x#1_2
   map#2_2
   main_2

(c) The following functions are completely defined:
none

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:

   plus_x#1(0, x8) -> x8 [1]
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14)) [1]
   map#2(plus_x(x2), Nil) -> Nil [1]
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1]
   main(x5, x12) -> map#2(plus_x(x12), x5) [1]

The TRS has the following type information:
plus_x#1 :: 0:S -> 0:S -> 0:S
0 :: 0:S
S :: 0:S -> 0:S
map#2 :: plus_x -> Nil:Cons -> Nil:Cons
plus_x :: 0:S -> plus_x
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> 0:S -> Nil:Cons
const :: plus_x

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:

   plus_x#1(0, x8) -> x8 [1]
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14)) [1]
   map#2(plus_x(x2), Nil) -> Nil [1]
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1]
   main(x5, x12) -> map#2(plus_x(x12), x5) [1]

The TRS has the following type information:
plus_x#1 :: 0:S -> 0:S -> 0:S
0 :: 0:S
S :: 0:S -> 0:S
map#2 :: plus_x -> Nil:Cons -> Nil:Cons
plus_x :: 0:S -> plus_x
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> 0:S -> Nil:Cons
const :: plus_x

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:

   main(z, z') -{ 1 }-> map#2(1 + x12, x5) :|: x5 >= 0, x12 >= 0, z = x5, z' = x12
   map#2(z, z') -{ 1 }-> 0 :|: z = 1 + x2, x2 >= 0, z' = 0
   map#2(z, z') -{ 1 }-> 1 + plus_x#1(x6, x4) + map#2(1 + x6, x2) :|: x4 >= 0, z = 1 + x6, z' = 1 + x4 + x2, x6 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> x8 :|: x8 >= 0, z = 0, z' = x8
   plus_x#1(z, z') -{ 1 }-> 1 + plus_x#1(x12, x14) :|: z = 1 + x12, x12 >= 0, x14 >= 0, z' = x14


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

(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:

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0


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

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

   { plus_x#1 }
   { map#2 }
   { main }

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

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

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus_x#1}, {map#2}, {main}

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

(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:

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus_x#1}, {map#2}, {main}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus_x#1}, {map#2}, {main}
Previous analysis results are:
  plus_x#1: runtime: ?, size: O(n^1) [z + z']

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus_x#1
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:

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 }-> 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {map#2}, {main}
Previous analysis results are:
  plus_x#1: runtime: O(n^1) [1 + z], size: O(n^1) [z + 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:

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 + z }-> 1 + s' + map#2(1 + (z - 1), x2) :|: s' >= 0, s' <= z - 1 + x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {map#2}, {main}
Previous analysis results are:
  plus_x#1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 + z }-> 1 + s' + map#2(1 + (z - 1), x2) :|: s' >= 0, s' <= z - 1 + x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {map#2}, {main}
Previous analysis results are:
  plus_x#1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  map#2: runtime: ?, size: O(n^2) [z*z' + z' + z'^2]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   main(z, z') -{ 1 }-> map#2(1 + z', z) :|: z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 1 + z }-> 1 + s' + map#2(1 + (z - 1), x2) :|: s' >= 0, s' <= z - 1 + x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {main}
Previous analysis results are:
  plus_x#1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  map#2: runtime: O(n^2) [1 + z*z' + z'], size: O(n^2) [z*z' + z' + z'^2]

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

(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:

   main(z, z') -{ 2 + 2*z + z*z' }-> s1 :|: s1 >= 0, s1 <= z * (1 + z') + z * z + z, z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 2 + x2 + x2*z + z }-> 1 + s' + s'' :|: s'' >= 0, s'' <= x2 * (1 + (z - 1)) + x2 * x2 + x2, s' >= 0, s' <= z - 1 + x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {main}
Previous analysis results are:
  plus_x#1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  map#2: runtime: O(n^2) [1 + z*z' + z'], size: O(n^2) [z*z' + z' + z'^2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   main(z, z') -{ 2 + 2*z + z*z' }-> s1 :|: s1 >= 0, s1 <= z * (1 + z') + z * z + z, z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 2 + x2 + x2*z + z }-> 1 + s' + s'' :|: s'' >= 0, s'' <= x2 * (1 + (z - 1)) + x2 * x2 + x2, s' >= 0, s' <= z - 1 + x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {main}
Previous analysis results are:
  plus_x#1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  map#2: runtime: O(n^2) [1 + z*z' + z'], size: O(n^2) [z*z' + z' + z'^2]
  main: runtime: ?, size: O(n^2) [2*z + z*z' + z^2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   main(z, z') -{ 2 + 2*z + z*z' }-> s1 :|: s1 >= 0, s1 <= z * (1 + z') + z * z + z, z >= 0, z' >= 0
   map#2(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   map#2(z, z') -{ 2 + x2 + x2*z + z }-> 1 + s' + s'' :|: s'' >= 0, s'' <= x2 * (1 + (z - 1)) + x2 * x2 + x2, s' >= 0, s' <= z - 1 + x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
   plus_x#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus_x#1(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  plus_x#1: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  map#2: runtime: O(n^2) [1 + z*z' + z'], size: O(n^2) [z*z' + z' + z'^2]
  main: runtime: O(n^2) [2 + 2*z + z*z'], size: O(n^2) [2*z + z*z' + z^2]

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

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

(34)
BOUNDS(1, n^2)

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

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

(36)
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^2).


The TRS R consists of the following rules:

   plus_x#1(0, x8) -> x8
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14))
   map#2(plus_x(x2), Nil) -> Nil
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
   main(x5, x12) -> map#2(plus_x(x12), x5)

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

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

The rewrite sequence

plus_x#1(S(x12), x14) ->^+ S(plus_x#1(x12, x14))

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

The pumping substitution is [x12 / S(x12)].

The result substitution is [ ].




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

(38)
Complex Obligation (BEST)

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

(39)
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^2).


The TRS R consists of the following rules:

   plus_x#1(0, x8) -> x8
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14))
   map#2(plus_x(x2), Nil) -> Nil
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
   main(x5, x12) -> map#2(plus_x(x12), x5)

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

(40) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(41)
BOUNDS(n^1, INF)

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

(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, n^2).


The TRS R consists of the following rules:

   plus_x#1(0, x8) -> x8
   plus_x#1(S(x12), x14) -> S(plus_x#1(x12, x14))
   map#2(plus_x(x2), Nil) -> Nil
   map#2(plus_x(x6), Cons(x4, x2)) -> Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
   main(x5, x12) -> map#2(plus_x(x12), x5)

S is empty.
Rewrite Strategy: INNERMOST