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

0 CpxRelTRS
↳1 DecreasingLoopProof (⇔, 1369 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 RewriteLemmaProof (LOWER BOUND(ID), 165 ms)
    ↳10 BEST
        ↳11 typed CpxTrs
            ↳12 RewriteLemmaProof (LOWER BOUND(ID), 216 ms)
            ↳13 BEST
                ↳14 typed CpxTrs
                    ↳15 NoRewriteLemmaProof (LOWER BOUND(ID), 11 ms)
                    ↳16 typed CpxTrs
                    ↳17 NoRewriteLemmaProof (LOWER BOUND(ID), 330 ms)
                    ↳18 typed CpxTrs
                    ↳19 LowerBoundsProof (⇔, 0 ms)
                    ↳20 BOUNDS(n^1, INF)
                ↳21 typed CpxTrs
                    ↳22 LowerBoundsProof (⇔, 0 ms)
                    ↳23 BOUNDS(n^1, INF)
        ↳24 typed CpxTrs
            ↳25 LowerBoundsProof (⇔, 0 ms)
            ↳26 BOUNDS(1, INF)

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)

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, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
merge(Cons(0, xs'), Cons(x, xs)) →+ Cons(0, merge(xs', Cons(x, xs)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs' / Cons(0, xs')].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)

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, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mergesort, splitmerge, merge, <=

They will be analysed ascendingly in the following order:
mergesort = splitmerge
merge < splitmerge
<= < merge

(8) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
<=, mergesort, splitmerge, merge

They will be analysed ascendingly in the following order:
mergesort = splitmerge
merge < splitmerge
<= < merge

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)

Induction Base:
<=(gen_S:0'5_0(0), gen_S:0'5_0(0)) →RΩ(0)
True

Induction Step:
<=(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(n7_0, 1))) →RΩ(0)
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) →IH
True

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n0).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
merge, mergesort, splitmerge

They will be analysed ascendingly in the following order:
mergesort = splitmerge
merge < splitmerge

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
merge(gen_Cons:Nil4_0(n282_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n282_0)), rt ∈ Ω(1 + n2820)

Induction Base:
merge(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(1)) →RΩ(1)
gen_Cons:Nil4_0(1)

Induction Step:
merge(gen_Cons:Nil4_0(+(n282_0, 1)), gen_Cons:Nil4_0(1)) →RΩ(1)
merge[Ite](<=(0', 0'), Cons(0', gen_Cons:Nil4_0(n282_0)), Cons(0', gen_Cons:Nil4_0(0))) →LΩ(0)
merge[Ite](True, Cons(0', gen_Cons:Nil4_0(n282_0)), Cons(0', gen_Cons:Nil4_0(0))) →RΩ(0)
Cons(0', merge(gen_Cons:Nil4_0(n282_0), Cons(0', gen_Cons:Nil4_0(0)))) →IH
Cons(0', gen_Cons:Nil4_0(+(1, c283_0)))

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)
merge(gen_Cons:Nil4_0(n282_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n282_0)), rt ∈ Ω(1 + n2820)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
splitmerge, mergesort

They will be analysed ascendingly in the following order:
mergesort = splitmerge

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol splitmerge.

(16) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)
merge(gen_Cons:Nil4_0(n282_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n282_0)), rt ∈ Ω(1 + n2820)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
mergesort

They will be analysed ascendingly in the following order:
mergesort = splitmerge

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol mergesort.

(18) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)
merge(gen_Cons:Nil4_0(n282_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n282_0)), rt ∈ Ω(1 + n2820)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
merge(gen_Cons:Nil4_0(n282_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n282_0)), rt ∈ Ω(1 + n2820)

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)
merge(gen_Cons:Nil4_0(n282_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n282_0)), rt ∈ Ω(1 + n2820)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
merge(gen_Cons:Nil4_0(n282_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n282_0)), rt ∈ Ω(1 + n2820)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) → splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) → Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) → merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) → Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) → splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) → merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) → Nil
merge(Nil, xs2) → xs2
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → mergesort(xs)
<=(S(x), S(y)) → <=(x, y)
<=(0', y) → True
<=(S(x), 0') → False
merge[Ite](False, xs1, Cons(x, xs)) → Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) → Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
splitmerge :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil → Cons:Nil → Cons:Nil
merge[Ite] :: True:False → Cons:Nil → Cons:Nil → Cons:Nil
<= :: S:0' → S:0' → True:False
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → True, rt ∈ Ω(0)

(26) BOUNDS(1, INF)