WORST_CASE(?,O(n^1))
proof of input_XGTcvmtOF7.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 121 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), 409 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 131 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 243 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
(34) CpxRNTS
(35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 702 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
(40) CpxRNTS
(41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 242 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
(46) CpxRNTS
(47) FinalProof [FINISHED, 0 ms]
(48) 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:

   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs)
   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs)
   foldr(a, Cons(x, xs)) -> op(x, foldr(a, xs))
   foldr(a, Nil) -> a
   foldl(a, Nil) -> a
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   op(x, S(0)) -> S(x)
   op(S(0), y) -> S(y)
   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), 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:

   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs) [1]
   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs) [1]
   foldr(a, Cons(x, xs)) -> op(x, foldr(a, xs)) [1]
   foldr(a, Nil) -> a [1]
   foldl(a, Nil) -> a [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   op(x, S(0)) -> S(x) [1]
   op(S(0), y) -> S(y) [1]
   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), 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:

   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs) [1]
   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs) [1]
   foldr(a, Cons(x, xs)) -> op(x, foldr(a, xs)) [1]
   foldr(a, Nil) -> a [1]
   foldl(a, Nil) -> a [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   op(x, S(0)) -> S(x) [1]
   op(S(0), y) -> S(y) [1]
   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), Nil)) [1]

The TRS has the following type information:
foldl :: 0:S -> Cons:Nil -> 0:S
Cons :: 0:S -> Cons:Nil -> Cons:Nil
S :: 0:S -> 0:S
0 :: 0:S
foldr :: 0:S -> Cons:Nil -> 0:S
op :: 0:S -> 0:S -> 0:S
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0:S -> Cons:Nil -> Cons: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:

   foldl_2
   notEmpty_1
   fold_2

(c) The following functions are completely defined:

   foldr_2
   op_2

Due to the following rules being added:

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

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:

   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs) [1]
   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs) [1]
   foldr(a, Cons(x, xs)) -> op(x, foldr(a, xs)) [1]
   foldr(a, Nil) -> a [1]
   foldl(a, Nil) -> a [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   op(x, S(0)) -> S(x) [1]
   op(S(0), y) -> S(y) [1]
   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), Nil)) [1]
   op(v0, v1) -> 0 [0]

The TRS has the following type information:
foldl :: 0:S -> Cons:Nil -> 0:S
Cons :: 0:S -> Cons:Nil -> Cons:Nil
S :: 0:S -> 0:S
0 :: 0:S
foldr :: 0:S -> Cons:Nil -> 0:S
op :: 0:S -> 0:S -> 0:S
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0:S -> Cons:Nil -> Cons: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:

   foldl(x, Cons(S(0), xs)) -> foldl(S(x), xs) [1]
   foldl(S(0), Cons(x, xs)) -> foldl(S(x), xs) [1]
   foldr(a, Cons(x, Cons(x', xs'))) -> op(x, op(x', foldr(a, xs'))) [2]
   foldr(a, Cons(x, Nil)) -> op(x, a) [2]
   foldr(a, Nil) -> a [1]
   foldl(a, Nil) -> a [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   op(x, S(0)) -> S(x) [1]
   op(S(0), y) -> S(y) [1]
   fold(a, xs) -> Cons(foldl(a, xs), Cons(foldr(a, xs), Nil)) [1]
   op(v0, v1) -> 0 [0]

The TRS has the following type information:
foldl :: 0:S -> Cons:Nil -> 0:S
Cons :: 0:S -> Cons:Nil -> Cons:Nil
S :: 0:S -> 0:S
0 :: 0:S
foldr :: 0:S -> Cons:Nil -> 0:S
op :: 0:S -> 0:S -> 0:S
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
fold :: 0:S -> Cons:Nil -> Cons: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:

   0 => 0
   Nil => 0
   True => 1
   False => 0

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

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

   fold(z, z') -{ 1 }-> 1 + foldl(a, xs) + (1 + foldr(a, xs) + 0) :|: z = a, xs >= 0, a >= 0, z' = xs
   foldl(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: z' = 1 + (1 + 0) + xs, xs >= 0, x >= 0, z = x
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldr(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, a) :|: z = a, a >= 0, x >= 0, z' = 1 + x + 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(a, xs'))) :|: z = a, a >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   op(z, z') -{ 1 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   op(z, z') -{ 1 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y


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

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

   op(z, z') -{ 1 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   op(z, z') -{ 1 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y
   op(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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

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

   fold(z, z') -{ 1 }-> 1 + foldl(a, xs) + (1 + foldr(a, xs) + 0) :|: z = a, xs >= 0, a >= 0, z' = xs
   foldl(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: z' = 1 + (1 + 0) + xs, xs >= 0, x >= 0, z = x
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldr(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(a, xs'))) :|: z = a, a >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z = a, a >= 0, x >= 0, z' = 1 + x + 0, v0 >= 0, v1 >= 0, x = v0, a = v1
   foldr(z, z') -{ 3 }-> 1 + x' :|: z = a, a >= 0, x >= 0, z' = 1 + x + 0, x' >= 0, a = 1 + 0, x = x'
   foldr(z, z') -{ 3 }-> 1 + y :|: z = a, a >= 0, x >= 0, z' = 1 + x + 0, x = 1 + 0, y >= 0, a = y
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   op(z, z') -{ 1 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   op(z, z') -{ 1 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y


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

(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:

   fold(z, z') -{ 1 }-> 1 + foldl(z, z') + (1 + foldr(z, z') + 0) :|: z' >= 0, z >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> foldl(1 + z, z' - 2) :|: z' - 2 >= 0, z >= 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0


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

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

   { foldl }
   { notEmpty }
   { op }
   { foldr }
   { fold }

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

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

   fold(z, z') -{ 1 }-> 1 + foldl(z, z') + (1 + foldr(z, z') + 0) :|: z' >= 0, z >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> foldl(1 + z, z' - 2) :|: z' - 2 >= 0, z >= 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {foldl}, {notEmpty}, {op}, {foldr}, {fold}

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

(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:

   fold(z, z') -{ 1 }-> 1 + foldl(z, z') + (1 + foldr(z, z') + 0) :|: z' >= 0, z >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> foldl(1 + z, z' - 2) :|: z' - 2 >= 0, z >= 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {foldl}, {notEmpty}, {op}, {foldr}, {fold}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 1 }-> 1 + foldl(z, z') + (1 + foldr(z, z') + 0) :|: z' >= 0, z >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> foldl(1 + z, z' - 2) :|: z' - 2 >= 0, z >= 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {foldl}, {notEmpty}, {op}, {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: ?, size: O(n^1) [z + z']

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 1 }-> 1 + foldl(z, z') + (1 + foldr(z, z') + 0) :|: z' >= 0, z >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldl(z, z') -{ 1 }-> foldl(1 + x, xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> foldl(1 + z, z' - 2) :|: z' - 2 >= 0, z >= 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {op}, {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']

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

(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:

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {op}, {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {notEmpty}, {op}, {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: ?, size: O(1) [1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {op}, {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]

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

(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:

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {op}, {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {op}, {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: ?, size: O(n^1) [z + z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: op
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:

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: O(1) [1], size: O(n^1) [z + 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:

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: O(1) [1], size: O(n^1) [z + z']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {foldr}, {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: O(1) [1], size: O(n^1) [z + z']
  foldr: runtime: ?, size: O(n^1) [z + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: foldr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 9 + 4*z'

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

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

   fold(z, z') -{ 2 + z' }-> 1 + s'' + (1 + foldr(z, z') + 0) :|: s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> op(x, op(x', foldr(z, xs'))) :|: z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: O(1) [1], size: O(n^1) [z + z']
  foldr: runtime: O(n^1) [9 + 4*z'], size: O(n^1) [z + z']

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

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

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

   fold(z, z') -{ 11 + 5*z' }-> 1 + s'' + (1 + s1 + 0) :|: s1 >= 0, s1 <= z + z', s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 13 + 4*xs' }-> s4 :|: s2 >= 0, s2 <= z + xs', s3 >= 0, s3 <= x' + s2, s4 >= 0, s4 <= x + s3, z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: O(1) [1], size: O(n^1) [z + z']
  foldr: runtime: O(n^1) [9 + 4*z'], size: O(n^1) [z + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 11 + 5*z' }-> 1 + s'' + (1 + s1 + 0) :|: s1 >= 0, s1 <= z + z', s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 13 + 4*xs' }-> s4 :|: s2 >= 0, s2 <= z + xs', s3 >= 0, s3 <= x' + s2, s4 >= 0, s4 <= x + s3, z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {fold}
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: O(1) [1], size: O(n^1) [z + z']
  foldr: runtime: O(n^1) [9 + 4*z'], size: O(n^1) [z + z']
  fold: runtime: ?, size: O(n^1) [2 + 2*z + 2*z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fold(z, z') -{ 11 + 5*z' }-> 1 + s'' + (1 + s1 + 0) :|: s1 >= 0, s1 <= z + z', s'' >= 0, s'' <= z + z', z' >= 0, z >= 0
   foldl(z, z') -{ z' }-> s :|: s >= 0, s <= 1 + z + (z' - 2), z' - 2 >= 0, z >= 0
   foldl(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= 1 + x + xs, xs >= 0, z' = 1 + x + xs, z = 1 + 0, x >= 0
   foldl(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 13 + 4*xs' }-> s4 :|: s2 >= 0, s2 <= z + xs', s3 >= 0, s3 <= x' + s2, s4 >= 0, s4 <= x + s3, z >= 0, x >= 0, x' >= 0, xs' >= 0, z' = 1 + x + (1 + x' + xs')
   foldr(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   foldr(z, z') -{ 2 }-> 0 :|: z >= 0, z' - 1 >= 0, v0 >= 0, z' - 1 = v0
   foldr(z, z') -{ 3 }-> 1 + x' :|: z >= 0, z' - 1 >= 0, x' >= 0, z = 1 + 0, z' - 1 = x'
   foldr(z, z') -{ 3 }-> 1 + z :|: z >= 0, z' - 1 >= 0, z' - 1 = 1 + 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 0 :|: z = 0
   op(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   op(z, z') -{ 1 }-> 1 + z :|: z >= 0, z' = 1 + 0
   op(z, z') -{ 1 }-> 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  foldl: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  notEmpty: runtime: O(1) [1], size: O(1) [1]
  op: runtime: O(1) [1], size: O(n^1) [z + z']
  foldr: runtime: O(n^1) [9 + 4*z'], size: O(n^1) [z + z']
  fold: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [2 + 2*z + 2*z']

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(47) FinalProof (FINISHED)
Computed overall runtime complexity
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(48)
BOUNDS(1, n^1)