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

(0) Obligation:

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

merge(nil, y) → y
merge(x, nil) → x
merge(.(x, y), .(u, v)) → if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))
if(true, x, y) → x
if(false, x, y) → x

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(nil, y) → y
merge(x, nil) → x
merge(.(x, y), .(u, v)) → if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))
if(true, x, y) → x
if(false, x, y) → x

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
./0
</0
</1

(4) Obligation:

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

merge(nil, y) → y
merge(x, nil) → x
merge(.(y), .(v)) → if(<, .(merge(y, .(v))), .(merge(.(y), v)))
++(nil, y) → y
++(.(y), z) → .(++(y, z))
if(true, x, y) → x
if(false, x, y) → x

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
merge(nil, y) → y
merge(x, nil) → x
merge(.(y), .(v)) → if(<, .(merge(y, .(v))), .(merge(.(y), v)))
++(nil, y) → y
++(.(y), z) → .(++(y, z))
if(true, x, y) → x
if(false, x, y) → x

Types:
merge :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
if :: <:true:false → nil:. → nil:. → nil:.
< :: <:true:false
++ :: nil:. → nil:. → nil:.
true :: <:true:false
false :: <:true:false
hole_nil:.1_0 :: nil:.
hole_<:true:false2_0 :: <:true:false
gen_nil:.3_0 :: Nat → nil:.

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

Heuristically decided to analyse the following defined symbols:
merge, ++

(8) Obligation:

Innermost TRS:
Rules:
merge(nil, y) → y
merge(x, nil) → x
merge(.(y), .(v)) → if(<, .(merge(y, .(v))), .(merge(.(y), v)))
++(nil, y) → y
++(.(y), z) → .(++(y, z))
if(true, x, y) → x
if(false, x, y) → x

Types:
merge :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
if :: <:true:false → nil:. → nil:. → nil:.
< :: <:true:false
++ :: nil:. → nil:. → nil:.
true :: <:true:false
false :: <:true:false
hole_nil:.1_0 :: nil:.
hole_<:true:false2_0 :: <:true:false
gen_nil:.3_0 :: Nat → nil:.

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))

The following defined symbols remain to be analysed:
merge, ++

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

Proved the following rewrite lemma:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
merge(gen_nil:.3_0(+(1, 0)), gen_nil:.3_0(1))

Induction Step:
merge(gen_nil:.3_0(+(1, +(n5_0, 1))), gen_nil:.3_0(1)) →_R^Ω(1)
if(<, .(merge(gen_nil:.3_0(+(1, n5_0)), .(gen_nil:.3_0(0)))), .(merge(.(gen_nil:.3_0(+(1, n5_0))), gen_nil:.3_0(0)))) →_IH
if(<, .(*4_0), .(merge(.(gen_nil:.3_0(+(1, n5_0))), gen_nil:.3_0(0)))) →_R^Ω(1)
if(<, .(*4_0), .(.(gen_nil:.3_0(+(1, n5_0)))))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
merge(nil, y) → y
merge(x, nil) → x
merge(.(y), .(v)) → if(<, .(merge(y, .(v))), .(merge(.(y), v)))
++(nil, y) → y
++(.(y), z) → .(++(y, z))
if(true, x, y) → x
if(false, x, y) → x

Types:
merge :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
if :: <:true:false → nil:. → nil:. → nil:.
< :: <:true:false
++ :: nil:. → nil:. → nil:.
true :: <:true:false
false :: <:true:false
hole_nil:.1_0 :: nil:.
hole_<:true:false2_0 :: <:true:false
gen_nil:.3_0 :: Nat → nil:.

Lemmas:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))

The following defined symbols remain to be analysed:
++

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

Proved the following rewrite lemma:
++(gen_nil:.3_0(n7751_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n7751_0, b)), rt ∈ Ω(1 + n7751₀)

Induction Base:
++(gen_nil:.3_0(0), gen_nil:.3_0(b)) →_R^Ω(1)
gen_nil:.3_0(b)

Induction Step:
++(gen_nil:.3_0(+(n7751_0, 1)), gen_nil:.3_0(b)) →_R^Ω(1)
.(++(gen_nil:.3_0(n7751_0), gen_nil:.3_0(b))) →_IH
.(gen_nil:.3_0(+(b, c7752_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
merge(nil, y) → y
merge(x, nil) → x
merge(.(y), .(v)) → if(<, .(merge(y, .(v))), .(merge(.(y), v)))
++(nil, y) → y
++(.(y), z) → .(++(y, z))
if(true, x, y) → x
if(false, x, y) → x

Types:
merge :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
if :: <:true:false → nil:. → nil:. → nil:.
< :: <:true:false
++ :: nil:. → nil:. → nil:.
true :: <:true:false
false :: <:true:false
hole_nil:.1_0 :: nil:.
hole_<:true:false2_0 :: <:true:false
gen_nil:.3_0 :: Nat → nil:.

Lemmas:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)
++(gen_nil:.3_0(n7751_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n7751_0, b)), rt ∈ Ω(1 + n7751₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
merge(nil, y) → y
merge(x, nil) → x
merge(.(y), .(v)) → if(<, .(merge(y, .(v))), .(merge(.(y), v)))
++(nil, y) → y
++(.(y), z) → .(++(y, z))
if(true, x, y) → x
if(false, x, y) → x

Types:
merge :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
if :: <:true:false → nil:. → nil:. → nil:.
< :: <:true:false
++ :: nil:. → nil:. → nil:.
true :: <:true:false
false :: <:true:false
hole_nil:.1_0 :: nil:.
hole_<:true:false2_0 :: <:true:false
gen_nil:.3_0 :: Nat → nil:.

Lemmas:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)
++(gen_nil:.3_0(n7751_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n7751_0, b)), rt ∈ Ω(1 + n7751₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
merge(nil, y) → y
merge(x, nil) → x
merge(.(y), .(v)) → if(<, .(merge(y, .(v))), .(merge(.(y), v)))
++(nil, y) → y
++(.(y), z) → .(++(y, z))
if(true, x, y) → x
if(false, x, y) → x

Types:
merge :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
if :: <:true:false → nil:. → nil:. → nil:.
< :: <:true:false
++ :: nil:. → nil:. → nil:.
true :: <:true:false
false :: <:true:false
hole_nil:.1_0 :: nil:.
hole_<:true:false2_0 :: <:true:false
gen_nil:.3_0 :: Nat → nil:.

Lemmas:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
merge(gen_nil:.3_0(+(1, n5_0)), gen_nil:.3_0(1)) → *4_0, rt ∈ Ω(n5₀)

(22) BOUNDS(n^1, INF)