WORST_CASE(?,O(n^1))
proof of input_8oXXrNLpax.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) InliningProof [UPPER BOUND(ID), 92 ms]
(12) CpxRNTS
(13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 58 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 1315 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 397 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 150 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
(34) CpxRNTS
(35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
(40) CpxRNTS
(41) FinalProof [FINISHED, 0 ms]
(42) 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:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1)) [1]
   ifappend(l1, l2, true) -> l2 [1]
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2)) [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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1)) [1]
   ifappend(l1, l2, true) -> l2 [1]
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2)) [1]

The TRS has the following type information:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> 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:

   hd_1
   append_2
   ifappend_3

(c) The following functions are completely defined:

   is_empty_1
   tl_1

Due to the following rules being added:

   tl(v0) -> nil [0]

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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1)) [1]
   ifappend(l1, l2, true) -> l2 [1]
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2)) [1]
   tl(v0) -> nil [0]

The TRS has the following type information:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
const :: hd

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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(nil, l2) -> ifappend(nil, l2, true) [2]
   append(cons(x', l'), l2) -> ifappend(cons(x', l'), l2, false) [2]
   ifappend(l1, l2, true) -> l2 [1]
   ifappend(cons(x'', l''), l2, false) -> cons(hd(cons(x'', l'')), append(l'', l2)) [2]
   ifappend(l1, l2, false) -> cons(hd(l1), append(nil, l2)) [1]
   tl(v0) -> nil [0]

The TRS has the following type information:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
const :: hd

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
   true => 1
   false => 0
   const => 0

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

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

   append(z, z') -{ 2 }-> ifappend(0, l2, 1) :|: z' = l2, z = 0, l2 >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', l2, 0) :|: x' >= 0, l' >= 0, z' = l2, l2 >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> l2 :|: z = l1, z' = l2, l1 >= 0, l2 >= 0, z'' = 1
   ifappend(z, z', z'') -{ 1 }-> 1 + hd(l1) + append(0, l2) :|: z'' = 0, z = l1, z' = l2, l1 >= 0, l2 >= 0
   ifappend(z, z', z'') -{ 2 }-> 1 + hd(1 + x'' + l'') + append(l'', l2) :|: z'' = 0, l'' >= 0, z' = l2, x'' >= 0, l2 >= 0, z = 1 + x'' + l''
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(11) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 2 }-> ifappend(0, l2, 1) :|: z' = l2, z = 0, l2 >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', l2, 0) :|: x' >= 0, l' >= 0, z' = l2, l2 >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> l2 :|: z = l1, z' = l2, l1 >= 0, l2 >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', l2) :|: z'' = 0, l'' >= 0, z' = l2, x'' >= 0, l2 >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, l2) :|: z'' = 0, z = l1, z' = l2, l1 >= 0, l2 >= 0, x >= 0, l >= 0, l1 = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { is_empty }
   { ifappend, append }
   { tl }
   { hd }

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}

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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: ?, size: O(1) [1]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]

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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using KoAT for: append
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2 + 6*z + 4*z^2 + 2*z'

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: ?, size: O(n^2) [z + z^2 + z']
  append: runtime: ?, size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ifappend
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 11 + 5*z

Computed RUNTIME bound using CoFloCo for: append
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 13 + 5*z

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [11 + 5*z], size: O(n^2) [z + z^2 + z']
  append: runtime: O(n^1) [13 + 5*z], size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']

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

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

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

   append(z, z') -{ 13 }-> s :|: s >= 0, s <= 0 + 0 * 0 + z', z = 0, z' >= 0
   append(z, z') -{ 18 + 5*l' + 5*x' }-> s' :|: s' >= 0, s' <= 1 + x' + l' + (1 + x' + l') * (1 + x' + l') + z', x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 16 + 5*l'' }-> 1 + x + s'' :|: s'' >= 0, s'' <= 6 * l'' + 4 * (l'' * l'') + 2 * z' + 2, z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 15 }-> 1 + x + s1 :|: s1 >= 0, s1 <= 6 * 0 + 4 * (0 * 0) + 2 * z' + 2, z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [11 + 5*z], size: O(n^2) [z + z^2 + z']
  append: runtime: O(n^1) [13 + 5*z], size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 13 }-> s :|: s >= 0, s <= 0 + 0 * 0 + z', z = 0, z' >= 0
   append(z, z') -{ 18 + 5*l' + 5*x' }-> s' :|: s' >= 0, s' <= 1 + x' + l' + (1 + x' + l') * (1 + x' + l') + z', x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 16 + 5*l'' }-> 1 + x + s'' :|: s'' >= 0, s'' <= 6 * l'' + 4 * (l'' * l'') + 2 * z' + 2, z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 15 }-> 1 + x + s1 :|: s1 >= 0, s1 <= 6 * 0 + 4 * (0 * 0) + 2 * z' + 2, z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [11 + 5*z], size: O(n^2) [z + z^2 + z']
  append: runtime: O(n^1) [13 + 5*z], size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']
  tl: runtime: ?, size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 13 }-> s :|: s >= 0, s <= 0 + 0 * 0 + z', z = 0, z' >= 0
   append(z, z') -{ 18 + 5*l' + 5*x' }-> s' :|: s' >= 0, s' <= 1 + x' + l' + (1 + x' + l') * (1 + x' + l') + z', x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 16 + 5*l'' }-> 1 + x + s'' :|: s'' >= 0, s'' <= 6 * l'' + 4 * (l'' * l'') + 2 * z' + 2, z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 15 }-> 1 + x + s1 :|: s1 >= 0, s1 <= 6 * 0 + 4 * (0 * 0) + 2 * z' + 2, z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [11 + 5*z], size: O(n^2) [z + z^2 + z']
  append: runtime: O(n^1) [13 + 5*z], size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']
  tl: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   append(z, z') -{ 13 }-> s :|: s >= 0, s <= 0 + 0 * 0 + z', z = 0, z' >= 0
   append(z, z') -{ 18 + 5*l' + 5*x' }-> s' :|: s' >= 0, s' <= 1 + x' + l' + (1 + x' + l') * (1 + x' + l') + z', x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 16 + 5*l'' }-> 1 + x + s'' :|: s'' >= 0, s'' <= 6 * l'' + 4 * (l'' * l'') + 2 * z' + 2, z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 15 }-> 1 + x + s1 :|: s1 >= 0, s1 <= 6 * 0 + 4 * (0 * 0) + 2 * z' + 2, z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [11 + 5*z], size: O(n^2) [z + z^2 + z']
  append: runtime: O(n^1) [13 + 5*z], size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']
  tl: runtime: O(1) [1], size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 13 }-> s :|: s >= 0, s <= 0 + 0 * 0 + z', z = 0, z' >= 0
   append(z, z') -{ 18 + 5*l' + 5*x' }-> s' :|: s' >= 0, s' <= 1 + x' + l' + (1 + x' + l') * (1 + x' + l') + z', x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 16 + 5*l'' }-> 1 + x + s'' :|: s'' >= 0, s'' <= 6 * l'' + 4 * (l'' * l'') + 2 * z' + 2, z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 15 }-> 1 + x + s1 :|: s1 >= 0, s1 <= 6 * 0 + 4 * (0 * 0) + 2 * z' + 2, z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [11 + 5*z], size: O(n^2) [z + z^2 + z']
  append: runtime: O(n^1) [13 + 5*z], size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']
  tl: runtime: O(1) [1], size: O(n^1) [z]
  hd: runtime: ?, size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 13 }-> s :|: s >= 0, s <= 0 + 0 * 0 + z', z = 0, z' >= 0
   append(z, z') -{ 18 + 5*l' + 5*x' }-> s' :|: s' >= 0, s' <= 1 + x' + l' + (1 + x' + l') * (1 + x' + l') + z', x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 16 + 5*l'' }-> 1 + x + s'' :|: s'' >= 0, s'' <= 6 * l'' + 4 * (l'' * l'') + 2 * z' + 2, z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   ifappend(z, z', z'') -{ 15 }-> 1 + x + s1 :|: s1 >= 0, s1 <= 6 * 0 + 4 * (0 * 0) + 2 * z' + 2, z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [11 + 5*z], size: O(n^2) [z + z^2 + z']
  append: runtime: O(n^1) [13 + 5*z], size: O(n^2) [2 + 6*z + 4*z^2 + 2*z']
  tl: runtime: O(1) [1], size: O(n^1) [z]
  hd: runtime: O(1) [1], size: O(n^1) [z]

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

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

(42)
BOUNDS(1, n^1)