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:

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

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:

+'(*'(x, y), *'(x, z)) → *'(x, +'(y, z))
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) → +'(*'(x, +'(y, z)), u)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
+'(*'(x, y), *'(x, z)) → *'(x, +'(y, z))
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) → +'(*'(x, +'(y, z)), u)

Types:
+' :: *':u → *':u → *':u
*' :: a → *':u → *':u
u :: *':u
hole_*':u1_0 :: *':u
hole_a2_0 :: a
gen_*':u3_0 :: Nat → *':u

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

Heuristically decided to analyse the following defined symbols:
+'

(6) Obligation:

Innermost TRS:
Rules:
+'(*'(x, y), *'(x, z)) → *'(x, +'(y, z))
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) → +'(*'(x, +'(y, z)), u)

Types:
+' :: *':u → *':u → *':u
*' :: a → *':u → *':u
u :: *':u
hole_*':u1_0 :: *':u
hole_a2_0 :: a
gen_*':u3_0 :: Nat → *':u

Generator Equations:
gen_*':u3_0(0) ⇔ u
gen_*':u3_0(+(x, 1)) ⇔ *'(hole_a2_0, gen_*':u3_0(x))

The following defined symbols remain to be analysed:
+'

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
+'(gen_*':u3_0(+(1, 0)), gen_*':u3_0(+(1, 0)))

Induction Step:
+'(gen_*':u3_0(+(1, +(n5_0, 1))), gen_*':u3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
*'(hole_a2_0, +'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0)))) →_IH
*'(hole_a2_0, *4_0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
+'(*'(x, y), *'(x, z)) → *'(x, +'(y, z))
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) → +'(*'(x, +'(y, z)), u)

Types:
+' :: *':u → *':u → *':u
*' :: a → *':u → *':u
u :: *':u
hole_*':u1_0 :: *':u
hole_a2_0 :: a
gen_*':u3_0 :: Nat → *':u

Lemmas:
+'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_*':u3_0(0) ⇔ u
gen_*':u3_0(+(x, 1)) ⇔ *'(hole_a2_0, gen_*':u3_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
+'(*'(x, y), *'(x, z)) → *'(x, +'(y, z))
+'(+'(x, y), z) → +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) → +'(*'(x, +'(y, z)), u)

Types:
+' :: *':u → *':u → *':u
*' :: a → *':u → *':u
u :: *':u
hole_*':u1_0 :: *':u
hole_a2_0 :: a
gen_*':u3_0 :: Nat → *':u

Lemmas:
+'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_*':u3_0(0) ⇔ u
gen_*':u3_0(+(x, 1)) ⇔ *'(hole_a2_0, gen_*':u3_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(14) BOUNDS(n^1, INF)