WORST_CASE(?,O(n^1))
proof of input_3evJRlrxji.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 218 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxWeightedTrs
(5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedTrs
(9) CompletionProof [UPPER BOUND(ID), 0 ms]
(10) CpxTypedWeightedCompleteTrs
(11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 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), 0 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 617 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 69 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 293 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 411 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 591 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 428 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
(48) CpxRNTS
(49) FinalProof [FINISHED, 0 ms]
(50) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   map(Cons(x, xs)) -> Cons(f(x), map(xs))
   map(Nil) -> Nil
   goal(xs) -> map(xs)
   f(x) -> *(x, x)
   +Full(S(x), y) -> +Full(x, S(y))
   +Full(0, y) -> y

The (relative) TRS S consists of the following rules:

   *(x, S(S(y))) -> +(x, *(x, S(y)))
   *(x, S(0)) -> x
   *(x, 0) -> 0
   *(0, y) -> 0

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   map(Cons(x, xs)) -> Cons(f(x), map(xs))
   map(Nil) -> Nil
   goal(xs) -> map(xs)
   f(x) -> *(x, x)
   +Full(S(x), y) -> +Full(x, S(y))
   +Full(0, y) -> y

The (relative) TRS S consists of the following rules:

   *(x, S(S(y))) -> +(x, *(x, S(y)))
   *(x, S(0)) -> x
   *(x, 0) -> 0
   *(0, y) -> 0

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (UPPER BOUND(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:

   map(Cons(x, xs)) -> Cons(f(x), map(xs)) [1]
   map(Nil) -> Nil [1]
   goal(xs) -> map(xs) [1]
   f(x) -> *(x, x) [1]
   +Full(S(x), y) -> +Full(x, S(y)) [1]
   +Full(0, y) -> y [1]
   *(x, S(S(y))) -> +(x, *(x, S(y))) [0]
   *(x, S(0)) -> x [0]
   *(x, 0) -> 0 [0]
   *(0, y) -> 0 [0]

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

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

* => times

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

(6)
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:

   map(Cons(x, xs)) -> Cons(f(x), map(xs)) [1]
   map(Nil) -> Nil [1]
   goal(xs) -> map(xs) [1]
   f(x) -> times(x, x) [1]
   +Full(S(x), y) -> +Full(x, S(y)) [1]
   +Full(0, y) -> y [1]
   times(x, S(S(y))) -> +(x, times(x, S(y))) [0]
   times(x, S(0)) -> x [0]
   times(x, 0) -> 0 [0]
   times(0, y) -> 0 [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   map(Cons(x, xs)) -> Cons(f(x), map(xs)) [1]
   map(Nil) -> Nil [1]
   goal(xs) -> map(xs) [1]
   f(x) -> times(x, x) [1]
   +Full(S(x), y) -> +Full(x, S(y)) [1]
   +Full(0, y) -> y [1]
   times(x, S(S(y))) -> +(x, times(x, S(y))) [0]
   times(x, S(0)) -> x [0]
   times(x, 0) -> 0 [0]
   times(0, y) -> 0 [0]

The TRS has the following type information:
map :: Cons:Nil -> Cons:Nil
Cons :: S:0:+ -> Cons:Nil -> Cons:Nil
f :: S:0:+ -> S:0:+
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
times :: S:0:+ -> S:0:+ -> S:0:+
+Full :: S:0:+ -> S:0:+ -> S:0:+
S :: S:0:+ -> S:0:+
0 :: S:0:+
+ :: S:0:+ -> S:0:+ -> S:0:+

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

(9) 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:

   map_1
   goal_1
   f_1
   +Full_2

(c) The following functions are completely defined:

   times_2

Due to the following rules being added:

   times(v0, v1) -> 0 [0]

And the following fresh constants: none

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

(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:

   map(Cons(x, xs)) -> Cons(f(x), map(xs)) [1]
   map(Nil) -> Nil [1]
   goal(xs) -> map(xs) [1]
   f(x) -> times(x, x) [1]
   +Full(S(x), y) -> +Full(x, S(y)) [1]
   +Full(0, y) -> y [1]
   times(x, S(S(y))) -> +(x, times(x, S(y))) [0]
   times(x, S(0)) -> x [0]
   times(x, 0) -> 0 [0]
   times(0, y) -> 0 [0]
   times(v0, v1) -> 0 [0]

The TRS has the following type information:
map :: Cons:Nil -> Cons:Nil
Cons :: S:0:+ -> Cons:Nil -> Cons:Nil
f :: S:0:+ -> S:0:+
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
times :: S:0:+ -> S:0:+ -> S:0:+
+Full :: S:0:+ -> S:0:+ -> S:0:+
S :: S:0:+ -> S:0:+
0 :: S:0:+
+ :: S:0:+ -> S:0:+ -> S:0:+

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

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

(12)
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:

   map(Cons(x, xs)) -> Cons(f(x), map(xs)) [1]
   map(Nil) -> Nil [1]
   goal(xs) -> map(xs) [1]
   f(x) -> times(x, x) [1]
   +Full(S(x), y) -> +Full(x, S(y)) [1]
   +Full(0, y) -> y [1]
   times(x, S(S(y))) -> +(x, times(x, S(y))) [0]
   times(x, S(0)) -> x [0]
   times(x, 0) -> 0 [0]
   times(0, y) -> 0 [0]
   times(v0, v1) -> 0 [0]

The TRS has the following type information:
map :: Cons:Nil -> Cons:Nil
Cons :: S:0:+ -> Cons:Nil -> Cons:Nil
f :: S:0:+ -> S:0:+
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
times :: S:0:+ -> S:0:+ -> S:0:+
+Full :: S:0:+ -> S:0:+ -> S:0:+
S :: S:0:+ -> S:0:+
0 :: S:0:+
+ :: S:0:+ -> S:0:+ -> S:0:+

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

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   0 => 0

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

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

   +Full(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   +Full(z, z') -{ 1 }-> +Full(x, 1 + y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   f(z) -{ 1 }-> times(x, x) :|: x >= 0, z = x
   goal(z) -{ 1 }-> map(xs) :|: xs >= 0, z = xs
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> x :|: x >= 0, z' = 1 + 0, z = x
   times(z, z') -{ 0 }-> 0 :|: x >= 0, z = x, z' = 0
   times(z, z') -{ 0 }-> 0 :|: y >= 0, z = 0, z' = y
   times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   times(z, z') -{ 0 }-> 1 + x + times(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 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:

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> times(z, z) :|: z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + times(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0


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

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

   { times }
   { +Full }
   { f }
   { map }
   { goal }

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

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

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> times(z, z) :|: z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + times(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {times}, {+Full}, {f}, {map}, {goal}

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

(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:

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> times(z, z) :|: z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + times(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {times}, {+Full}, {f}, {map}, {goal}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> times(z, z) :|: z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + times(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {times}, {+Full}, {f}, {map}, {goal}
Previous analysis results are:
  times: runtime: ?, size: O(n^2) [z + z*z' + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> times(z, z) :|: z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + times(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {+Full}, {f}, {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + 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:

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {+Full}, {f}, {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {+Full}, {f}, {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: ?, size: O(n^1) [z + z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   +Full(z, z') -{ 1 }-> +Full(z - 1, 1 + z') :|: z - 1 >= 0, z' >= 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {f}, {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

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

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

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {f}, {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {f}, {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: ?, size: O(n^2) [2*z + z^2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 1 }-> 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: O(1) [1], size: O(n^2) [2*z + z^2]

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

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

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

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 2 }-> 1 + s1 + map(xs) :|: s1 >= 0, s1 <= 2 * x + x * x, z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: O(1) [1], size: O(n^2) [2*z + z^2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 2 }-> 1 + s1 + map(xs) :|: s1 >= 0, s1 <= 2 * x + x * x, z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {map}, {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: O(1) [1], size: O(n^2) [2*z + z^2]
  map: runtime: ?, size: INF

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 1 }-> map(z) :|: z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 2 }-> 1 + s1 + map(xs) :|: s1 >= 0, s1 <= 2 * x + x * x, z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: O(1) [1], size: O(n^2) [2*z + z^2]
  map: runtime: O(n^1) [1 + 2*z], size: INF

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

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

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

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 2 + 2*z }-> s3 :|: s3 >= 0, s3 <= inf', z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 3 + 2*xs }-> 1 + s1 + s2 :|: s2 >= 0, s2 <= inf, s1 >= 0, s1 <= 2 * x + x * x, z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: O(1) [1], size: O(n^2) [2*z + z^2]
  map: runtime: O(n^1) [1 + 2*z], size: INF

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 2 + 2*z }-> s3 :|: s3 >= 0, s3 <= inf', z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 3 + 2*xs }-> 1 + s1 + s2 :|: s2 >= 0, s2 <= inf, s1 >= 0, s1 <= 2 * x + x * x, z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: O(1) [1], size: O(n^2) [2*z + z^2]
  map: runtime: O(n^1) [1 + 2*z], size: INF
  goal: runtime: ?, size: INF

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   +Full(z, z') -{ 1 + z }-> s'' :|: s'' >= 0, s'' <= z - 1 + (1 + z'), z - 1 >= 0, z' >= 0
   +Full(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   f(z) -{ 1 }-> s :|: s >= 0, s <= z * z + z + z, z >= 0
   goal(z) -{ 2 + 2*z }-> s3 :|: s3 >= 0, s3 <= inf', z >= 0
   map(z) -{ 1 }-> 0 :|: z = 0
   map(z) -{ 3 + 2*xs }-> 1 + s1 + s2 :|: s2 >= 0, s2 <= inf, s1 >= 0, s1 <= 2 * x + x * x, z = 1 + x + xs, xs >= 0, x >= 0
   times(z, z') -{ 0 }-> z :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z + s' :|: s' >= 0, s' <= (1 + (z' - 2)) * z + (1 + (z' - 2)) + z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z' + z']
  +Full: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  f: runtime: O(1) [1], size: O(n^2) [2*z + z^2]
  map: runtime: O(n^1) [1 + 2*z], size: INF
  goal: runtime: O(n^1) [2 + 2*z], size: INF

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

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

(50)
BOUNDS(1, n^1)