WORST_CASE(?,O(n^1))
proof of input_Zpk4DoQP4c.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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), 565 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 185 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 200 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 118 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
(32) CpxRNTS
(33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 209 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
(38) CpxRNTS
(39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 241 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
(44) CpxRNTS
(45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
(48) CpxRNTS
(49) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
(50) CpxRNTS
(51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(52) CpxRNTS
(53) IntTrsBoundProof [UPPER BOUND(ID), 138 ms]
(54) CpxRNTS
(55) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
(56) CpxRNTS
(57) FinalProof [FINISHED, 0 ms]
(58) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   append(@l1, @l2) -> append#1(@l1, @l2)
   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2))
   append#1(nil, @l2) -> @l2
   appendAll(@l) -> appendAll#1(@l)
   appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls))
   appendAll#1(nil) -> nil
   appendAll2(@l) -> appendAll2#1(@l)
   appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls))
   appendAll2#1(nil) -> nil
   appendAll3(@l) -> appendAll3#1(@l)
   appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls))
   appendAll3#1(nil) -> nil

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   append(@l1, @l2) -> append#1(@l1, @l2) [1]
   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1]
   append#1(nil, @l2) -> @l2 [1]
   appendAll(@l) -> appendAll#1(@l) [1]
   appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls)) [1]
   appendAll#1(nil) -> nil [1]
   appendAll2(@l) -> appendAll2#1(@l) [1]
   appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls)) [1]
   appendAll2#1(nil) -> nil [1]
   appendAll3(@l) -> appendAll3#1(@l) [1]
   appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls)) [1]
   appendAll3#1(nil) -> nil [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:

   append(@l1, @l2) -> append#1(@l1, @l2) [1]
   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1]
   append#1(nil, @l2) -> @l2 [1]
   appendAll(@l) -> appendAll#1(@l) [1]
   appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls)) [1]
   appendAll#1(nil) -> nil [1]
   appendAll2(@l) -> appendAll2#1(@l) [1]
   appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls)) [1]
   appendAll2#1(nil) -> nil [1]
   appendAll3(@l) -> appendAll3#1(@l) [1]
   appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls)) [1]
   appendAll3#1(nil) -> nil [1]

The TRS has the following type information:
append :: :::nil -> :::nil -> :::nil
append#1 :: :::nil -> :::nil -> :::nil
:: :: :::nil -> :::nil -> :::nil
nil :: :::nil
appendAll :: :::nil -> :::nil
appendAll#1 :: :::nil -> :::nil
appendAll2 :: :::nil -> :::nil
appendAll2#1 :: :::nil -> :::nil
appendAll3 :: :::nil -> :::nil
appendAll3#1 :: :::nil -> :::nil

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:
none

(c) The following functions are completely defined:

   appendAll2_1
   appendAll3_1
   appendAll_1
   appendAll#1_1
   appendAll2#1_1
   append_2
   appendAll3#1_1
   append#1_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   append(@l1, @l2) -> append#1(@l1, @l2) [1]
   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1]
   append#1(nil, @l2) -> @l2 [1]
   appendAll(@l) -> appendAll#1(@l) [1]
   appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll(@ls)) [1]
   appendAll#1(nil) -> nil [1]
   appendAll2(@l) -> appendAll2#1(@l) [1]
   appendAll2#1(::(@l1, @ls)) -> append(appendAll(@l1), appendAll2(@ls)) [1]
   appendAll2#1(nil) -> nil [1]
   appendAll3(@l) -> appendAll3#1(@l) [1]
   appendAll3#1(::(@l1, @ls)) -> append(appendAll2(@l1), appendAll3(@ls)) [1]
   appendAll3#1(nil) -> nil [1]

The TRS has the following type information:
append :: :::nil -> :::nil -> :::nil
append#1 :: :::nil -> :::nil -> :::nil
:: :: :::nil -> :::nil -> :::nil
nil :: :::nil
appendAll :: :::nil -> :::nil
appendAll#1 :: :::nil -> :::nil
appendAll2 :: :::nil -> :::nil
appendAll2#1 :: :::nil -> :::nil
appendAll3 :: :::nil -> :::nil
appendAll3#1 :: :::nil -> :::nil

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:

   append(@l1, @l2) -> append#1(@l1, @l2) [1]
   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2)) [1]
   append#1(nil, @l2) -> @l2 [1]
   appendAll(@l) -> appendAll#1(@l) [1]
   appendAll#1(::(@l1, @ls)) -> append(@l1, appendAll#1(@ls)) [2]
   appendAll#1(nil) -> nil [1]
   appendAll2(@l) -> appendAll2#1(@l) [1]
   appendAll2#1(::(@l1, @ls)) -> append(appendAll#1(@l1), appendAll2#1(@ls)) [3]
   appendAll2#1(nil) -> nil [1]
   appendAll3(@l) -> appendAll3#1(@l) [1]
   appendAll3#1(::(@l1, @ls)) -> append(appendAll2#1(@l1), appendAll3#1(@ls)) [3]
   appendAll3#1(nil) -> nil [1]

The TRS has the following type information:
append :: :::nil -> :::nil -> :::nil
append#1 :: :::nil -> :::nil -> :::nil
:: :: :::nil -> :::nil -> :::nil
nil :: :::nil
appendAll :: :::nil -> :::nil
appendAll#1 :: :::nil -> :::nil
appendAll2 :: :::nil -> :::nil
appendAll2#1 :: :::nil -> :::nil
appendAll3 :: :::nil -> :::nil
appendAll3#1 :: :::nil -> :::nil

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:

   nil => 0

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

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

   append(z, z') -{ 1 }-> append#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1
   append#1(z, z') -{ 1 }-> @l2 :|: z' = @l2, @l2 >= 0, z = 0
   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @l2) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(@l) :|: z = @l, @l >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(@l) :|: z = @l, @l >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(@l) :|: z = @l, @l >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 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:

   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0


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

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

   { append#1, append }
   { appendAll#1 }
   { appendAll }
   { appendAll2#1 }
   { appendAll3#1 }
   { appendAll2 }
   { appendAll3 }

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

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

   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {append#1,append}, {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}

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

(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:

   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {append#1,append}, {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: append
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:

   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {append#1,append}, {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: ?, size: O(n^1) [z + z']
  append: runtime: ?, size: O(n^1) [z + z']

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed RUNTIME bound using CoFloCo for: append
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 2*z

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

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

   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*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:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll#1}, {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: ?, size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: appendAll#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 6*z

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 1 }-> appendAll#1(z) :|: z >= 0
   appendAll#1(z) -{ 2 }-> append(@l1, appendAll#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 3 }-> append(appendAll#1(@l1), appendAll2#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [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:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll}, {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: appendAll
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 6*z

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]

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

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: appendAll2#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll2#1}, {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: appendAll2#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 8*z

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 1 }-> appendAll2#1(z) :|: z >= 0
   appendAll2#1(z) -{ 4 + 6*@l1 }-> append(s3, appendAll2#1(@ls)) :|: s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 3 }-> append(appendAll2#1(@l1), appendAll3#1(@ls)) :|: @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]

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

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 4 + 8*@l1 }-> append(s7, appendAll3#1(@ls)) :|: s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: appendAll3#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 4 + 8*@l1 }-> append(s7, appendAll3#1(@ls)) :|: s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll3#1}, {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: ?, size: O(n^1) [z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: appendAll3#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 10*z

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 1 }-> appendAll3#1(z) :|: z >= 0
   appendAll3#1(z) -{ 4 + 8*@l1 }-> append(s7, appendAll3#1(@ls)) :|: s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z]

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

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: appendAll2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll2}, {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z]
  appendAll2: runtime: ?, size: O(n^1) [z]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: appendAll2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 8*z

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z]
  appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z]

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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z]
  appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: appendAll3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: {appendAll3}
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z]
  appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z]
  appendAll3: runtime: ?, size: O(n^1) [z]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: appendAll3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 10*z

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
   append#1(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
   appendAll(z) -{ 2 + 6*z }-> s'' :|: s'' >= 0, s'' <= z, z >= 0
   appendAll#1(z) -{ 7 + 2*@l1 + 6*@ls }-> s2 :|: s1 >= 0, s1 <= @ls, s2 >= 0, s2 <= @l1 + s1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll2(z) -{ 2 + 8*z }-> s4 :|: s4 >= 0, s4 <= z, z >= 0
   appendAll2#1(z) -{ 9 + 6*@l1 + 8*@ls + 2*s3 }-> s6 :|: s5 >= 0, s5 <= @ls, s6 >= 0, s6 <= s3 + s5, s3 >= 0, s3 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll2#1(z) -{ 1 }-> 0 :|: z = 0
   appendAll3(z) -{ 2 + 10*z }-> s8 :|: s8 >= 0, s8 <= z, z >= 0
   appendAll3#1(z) -{ 9 + 8*@l1 + 10*@ls + 2*s7 }-> s10 :|: s9 >= 0, s9 <= @ls, s10 >= 0, s10 <= s7 + s9, s7 >= 0, s7 <= @l1, @ls >= 0, @l1 >= 0, z = 1 + @l1 + @ls
   appendAll3#1(z) -{ 1 }-> 0 :|: z = 0

Function symbols to be analyzed: 
Previous analysis results are:
  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  appendAll#1: runtime: O(n^1) [1 + 6*z], size: O(n^1) [z]
  appendAll: runtime: O(n^1) [2 + 6*z], size: O(n^1) [z]
  appendAll2#1: runtime: O(n^1) [1 + 8*z], size: O(n^1) [z]
  appendAll3#1: runtime: O(n^1) [1 + 10*z], size: O(n^1) [z]
  appendAll2: runtime: O(n^1) [2 + 8*z], size: O(n^1) [z]
  appendAll3: runtime: O(n^1) [2 + 10*z], size: O(n^1) [z]

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

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

(58)
BOUNDS(1, n^1)