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

(0) Obligation:

Runtime Complexity Relative TRS:
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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
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

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

Infered types.

(4) Obligation:

Innermost TRS:
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)
<=(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))

Types:
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
<= :: 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
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'

(5) OrderProof (LOWER BOUND(ID) transformation)

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

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

(6) Obligation:

Innermost TRS:
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)
<=(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))

Types:
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
<= :: 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
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:
<=, merge

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

(7) 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)) →_R^Ω(0)
True

Induction Step:
<=(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(n7_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n⁰).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
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)
<=(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))

Types:
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
<= :: 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
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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
merge(gen_Cons:Nil4_0(n240_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n240_0)), rt ∈ Ω(1 + n240₀)

Induction Base:
merge(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(1)) →_R^Ω(1)
gen_Cons:Nil4_0(1)

Induction Step:
merge(gen_Cons:Nil4_0(+(n240_0, 1)), gen_Cons:Nil4_0(1)) →_R^Ω(1)
merge[Ite](<=(0', 0'), Cons(0', gen_Cons:Nil4_0(n240_0)), Cons(0', gen_Cons:Nil4_0(0))) →_L^Ω(0)
merge[Ite](True, Cons(0', gen_Cons:Nil4_0(n240_0)), Cons(0', gen_Cons:Nil4_0(0))) →_R^Ω(0)
Cons(0', merge(gen_Cons:Nil4_0(n240_0), Cons(0', gen_Cons:Nil4_0(0)))) →_IH
Cons(0', gen_Cons:Nil4_0(+(1, c241_0)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
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)
<=(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))

Types:
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
<= :: 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
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(n240_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n240_0)), rt ∈ Ω(1 + n240₀)

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.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
merge(gen_Cons:Nil4_0(n240_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n240_0)), rt ∈ Ω(1 + n240₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
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)
<=(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))

Types:
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
<= :: 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
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(n240_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n240_0)), rt ∈ Ω(1 + n240₀)

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.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
merge(gen_Cons:Nil4_0(n240_0), gen_Cons:Nil4_0(1)) → gen_Cons:Nil4_0(+(1, n240_0)), rt ∈ Ω(1 + n240₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
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)
<=(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))

Types:
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
<= :: 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
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.

(19) 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)

(20) BOUNDS(1, INF)