WORST_CASE(?,O(n^1))
proof of input_8ezI11cnGS.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 202 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxWeightedTrs
(5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedTrs
(9) CompletionProof [UPPER BOUND(ID), 0 ms]
(10) CpxTypedWeightedCompleteTrs
(11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(14) CpxRNTS
(15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRNTS
(19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 428 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 80 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 3052 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 1025 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
(36) CpxRNTS
(37) FinalProof [FINISHED, 0 ms]
(38) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Nil, ys) -> ys
   goal(xs, ys) -> merge(xs, ys)

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

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

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

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

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


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Nil, ys) -> ys
   goal(xs, ys) -> merge(xs, ys)

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

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   <=(S(x), S(y)) -> <=(x, y) [0]
   <=(0, y) -> True [0]
   <=(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]

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

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

<= => lteq

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   lteq(S(x), S(y)) -> lteq(x, y) [0]
   lteq(0, y) -> True [0]
   lteq(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]

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

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

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

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   lteq(S(x), S(y)) -> lteq(x, y) [0]
   lteq(0, y) -> True [0]
   lteq(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]

The TRS has the following type information:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
lteq :: S:0 -> S:0 -> True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0 -> S:0
0 :: S:0
True :: True:False
False :: True:False

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   merge_2
   goal_2

(c) The following functions are completely defined:

   lteq_2
   merge[Ite]_3

Due to the following rules being added:

   lteq(v0, v1) -> null_lteq [0]
   merge[Ite](v0, v1, v2) -> Nil [0]

And the following fresh constants:    null_lteq

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   lteq(S(x), S(y)) -> lteq(x, y) [0]
   lteq(0, y) -> True [0]
   lteq(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]
   lteq(v0, v1) -> null_lteq [0]
   merge[Ite](v0, v1, v2) -> Nil [0]

The TRS has the following type information:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False:null_lteq -> Cons:Nil -> Cons:Nil -> Cons:Nil
lteq :: S:0 -> S:0 -> True:False:null_lteq
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0 -> S:0
0 :: S:0
True :: True:False:null_lteq
False :: True:False:null_lteq
null_lteq :: True:False:null_lteq

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(S(x''), xs'), Cons(S(y'), xs)) -> merge[Ite](lteq(x'', y'), Cons(S(x''), xs'), Cons(S(y'), xs)) [1]
   merge(Cons(0, xs'), Cons(x, xs)) -> merge[Ite](True, Cons(0, xs'), Cons(x, xs)) [1]
   merge(Cons(S(x1), xs'), Cons(0, xs)) -> merge[Ite](False, Cons(S(x1), xs'), Cons(0, xs)) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](null_lteq, Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   lteq(S(x), S(y)) -> lteq(x, y) [0]
   lteq(0, y) -> True [0]
   lteq(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]
   lteq(v0, v1) -> null_lteq [0]
   merge[Ite](v0, v1, v2) -> Nil [0]

The TRS has the following type information:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False:null_lteq -> Cons:Nil -> Cons:Nil -> Cons:Nil
lteq :: S:0 -> S:0 -> True:False:null_lteq
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0 -> S:0
0 :: S:0
True :: True:False:null_lteq
False :: True:False:null_lteq
null_lteq :: True:False:null_lteq

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

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

   Nil => 0
   0 => 0
   True => 2
   False => 1
   null_lteq => 0

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

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

   goal(z, z') -{ 1 }-> merge(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0
   lteq(z, z') -{ 0 }-> lteq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lteq(z, z') -{ 0 }-> 2 :|: y >= 0, z = 0, z' = y
   lteq(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   merge(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z = 1 + 0 + xs', xs' >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + xs) :|: xs >= 0, x1 >= 0, z' = 1 + 0 + xs, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, ys) :|: z = 2, xs >= 0, z' = 1 + x + xs, ys >= 0, x >= 0, z'' = ys
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs', xs) :|: xs >= 0, z = 1, xs' >= 0, x >= 0, z' = xs', z'' = 1 + x + xs


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

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs


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

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

   { lteq }
   { merge[Ite], merge }
   { goal }

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {lteq}, {merge[Ite],merge}, {goal}

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

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {lteq}, {merge[Ite],merge}, {goal}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: lteq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {lteq}, {merge[Ite],merge}, {goal}
Previous analysis results are:
  lteq: runtime: ?, size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> lteq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](lteq(x'', y'), 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {merge[Ite],merge}, {goal}
Previous analysis results are:
  lteq: runtime: O(1) [0], size: O(1) [2]

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

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {merge[Ite],merge}, {goal}
Previous analysis results are:
  lteq: runtime: O(1) [0], size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {merge[Ite],merge}, {goal}
Previous analysis results are:
  lteq: runtime: O(1) [0], size: O(1) [2]
  merge[Ite]: runtime: ?, size: O(n^1) [z' + z'']
  merge: runtime: ?, size: O(n^1) [z + z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: merge[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 8 + z' + z''

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

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

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

   goal(z, z') -{ 1 }-> merge(z, z') :|: z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](s, 1 + (1 + x'') + xs', 1 + (1 + y') + xs) :|: s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](2, 1 + 0 + (z - 1), 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 1 }-> merge[Ite](1, 1 + (1 + x1) + xs', 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 1 }-> merge[Ite](0, 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, z'') :|: z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs

Function symbols to be analyzed: {goal}
Previous analysis results are:
  lteq: runtime: O(1) [0], size: O(1) [2]
  merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z'']
  merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z']

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

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

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

   goal(z, z') -{ 10 + z + z' }-> s4 :|: s4 >= 0, s4 <= z + z', z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 13 + x'' + xs + xs' + y' }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + x'') + xs' + (1 + (1 + y') + xs), s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 10 + x + xs + z }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + (z - 1) + (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 11 + x1 + xs' + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + x1) + xs' + (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 11 + x + x' + xs + xs' }-> s3 :|: s3 >= 0, s3 <= 1 + x' + xs' + (1 + x + xs), xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 9 + xs + z' }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   merge[Ite](z, z', z'') -{ 9 + xs + z'' }-> 1 + x + s6 :|: s6 >= 0, s6 <= xs + z'', z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  lteq: runtime: O(1) [0], size: O(1) [2]
  merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z'']
  merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z, z') -{ 10 + z + z' }-> s4 :|: s4 >= 0, s4 <= z + z', z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 13 + x'' + xs + xs' + y' }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + x'') + xs' + (1 + (1 + y') + xs), s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 10 + x + xs + z }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + (z - 1) + (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 11 + x1 + xs' + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + x1) + xs' + (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 11 + x + x' + xs + xs' }-> s3 :|: s3 >= 0, s3 <= 1 + x' + xs' + (1 + x + xs), xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 9 + xs + z' }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   merge[Ite](z, z', z'') -{ 9 + xs + z'' }-> 1 + x + s6 :|: s6 >= 0, s6 <= xs + z'', z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  lteq: runtime: O(1) [0], size: O(1) [2]
  merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z'']
  merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z']
  goal: runtime: ?, size: O(n^1) [z + z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   goal(z, z') -{ 10 + z + z' }-> s4 :|: s4 >= 0, s4 <= z + z', z >= 0, z' >= 0
   lteq(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0
   lteq(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   lteq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   merge(z, z') -{ 13 + x'' + xs + xs' + y' }-> s'' :|: s'' >= 0, s'' <= 1 + (1 + x'') + xs' + (1 + (1 + y') + xs), s >= 0, s <= 2, z = 1 + (1 + x'') + xs', xs >= 0, z' = 1 + (1 + y') + xs, xs' >= 0, y' >= 0, x'' >= 0
   merge(z, z') -{ 10 + x + xs + z }-> s1 :|: s1 >= 0, s1 <= 1 + 0 + (z - 1) + (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
   merge(z, z') -{ 11 + x1 + xs' + z' }-> s2 :|: s2 >= 0, s2 <= 1 + (1 + x1) + xs' + (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
   merge(z, z') -{ 11 + x + x' + xs + xs' }-> s3 :|: s3 >= 0, s3 <= 1 + x' + xs' + (1 + x + xs), xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   merge[Ite](z, z', z'') -{ 9 + xs + z' }-> 1 + x + s5 :|: s5 >= 0, s5 <= z' + xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
   merge[Ite](z, z', z'') -{ 9 + xs + z'' }-> 1 + x + s6 :|: s6 >= 0, s6 <= xs + z'', z = 2, xs >= 0, z' = 1 + x + xs, z'' >= 0, x >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  lteq: runtime: O(1) [0], size: O(1) [2]
  merge[Ite]: runtime: O(n^1) [8 + z' + z''], size: O(n^1) [z' + z'']
  merge: runtime: O(n^1) [9 + z + z'], size: O(n^1) [z + z']
  goal: runtime: O(n^1) [10 + z + z'], size: O(n^1) [z + z']

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

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

(38)
BOUNDS(1, n^1)