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

(0) Obligation:

The Runtime Complexity (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: INNERMOST

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

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:

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

(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined symbols to avoid conflicts with arithmetic symbols:

<= => lteq

(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](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

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) 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

(7) CompletionProof (UPPER BOUND(ID) transformation)

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
goal

(c) The following functions are completely defined:

lteq
merge[Ite]

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

(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: 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

(9) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(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(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

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

(12) 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

(13) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(14) 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

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

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

(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

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

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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}
Previous analysis results are:

lteq: runtime: ?, size: O(1) [2]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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: {merge[Ite],merge}, {goal}
Previous analysis results are:

lteq: runtime: O(1) [0], size: O(1) [2]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(22) 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]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: merge[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z' + z''

Computed SIZE bound using CoFloCo for: merge
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z + z'

(24) 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(n1) [z' + z''] merge: runtime: ?, size: O(n1) [z + z']

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: merge[Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + z' + z''

Computed RUNTIME bound using CoFloCo for: merge
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 4 + z + z'

(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: {goal}
Previous analysis results are:

lteq: runtime: O(1) [0], size: O(1) [2] merge[Ite]: runtime: O(n1) [3 + z' + z''], size: O(n1) [z' + z''] merge: runtime: O(n1) [4 + z + z'], size: O(n1) [z + z']

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(28) Obligation:

Complexity RNTS consisting of the following rules:

goal(z, z') -{ 5 + z + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * 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') -{ 8 + x'' + xs + xs' + y' }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + (1 + x'') + xs') + 1 * (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') -{ 5 + x + xs + z }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + 0 + (z - 1)) + 1 * (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
merge(z, z') -{ 6 + x1 + xs' + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + x1) + xs') + 1 * (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
merge(z, z') -{ 6 + x + x' + xs + xs' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + x' + xs') + 1 * (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'') -{ 4 + xs + z' }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
merge[Ite](z, z', z'') -{ 4 + xs + z'' }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * xs + 1 * 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(n1) [3 + z' + z''], size: O(n1) [z' + z''] merge: runtime: O(n1) [4 + z + z'], size: O(n1) [z + z']

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z + z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

goal(z, z') -{ 5 + z + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * 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') -{ 8 + x'' + xs + xs' + y' }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + (1 + x'') + xs') + 1 * (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') -{ 5 + x + xs + z }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + 0 + (z - 1)) + 1 * (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
merge(z, z') -{ 6 + x1 + xs' + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + x1) + xs') + 1 * (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
merge(z, z') -{ 6 + x + x' + xs + xs' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + x' + xs') + 1 * (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'') -{ 4 + xs + z' }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
merge[Ite](z, z', z'') -{ 4 + xs + z'' }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * xs + 1 * 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(n1) [3 + z' + z''], size: O(n1) [z' + z''] merge: runtime: O(n1) [4 + z + z'], size: O(n1) [z + z'] goal: runtime: ?, size: O(n1) [z + z']

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 5 + z + z'

(32) Obligation:

Complexity RNTS consisting of the following rules:

goal(z, z') -{ 5 + z + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * 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') -{ 8 + x'' + xs + xs' + y' }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + (1 + x'') + xs') + 1 * (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') -{ 5 + x + xs + z }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + 0 + (z - 1)) + 1 * (1 + x + xs), xs >= 0, z' = 1 + x + xs, z - 1 >= 0, x >= 0
merge(z, z') -{ 6 + x1 + xs' + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + x1) + xs') + 1 * (1 + 0 + (z' - 1)), z' - 1 >= 0, x1 >= 0, z = 1 + (1 + x1) + xs', xs' >= 0
merge(z, z') -{ 6 + x + x' + xs + xs' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + x' + xs') + 1 * (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'') -{ 4 + xs + z' }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
merge[Ite](z, z', z'') -{ 4 + xs + z'' }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * xs + 1 * 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(n1) [3 + z' + z''], size: O(n1) [z' + z''] merge: runtime: O(n1) [4 + z + z'], size: O(n1) [z + z'] goal: runtime: O(n1) [5 + z + z'], size: O(n1) [z + z']

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^1)