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

Renamed function symbols to avoid clashes with predefined symbol.

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

Infered types.

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_*':u'1 :: *':u'
_hole_a2 :: a
_gen_*':u'3 :: Nat → *':u'

Heuristically decided to analyse the following defined symbols:
+'

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_*':u'1 :: *':u'
_hole_a2 :: a
_gen_*':u'3 :: Nat → *':u'

Generator Equations:
_gen_*':u'3(0) ⇔ u'
_gen_*':u'3(+(x, 1)) ⇔ *'(_hole_a2, _gen_*':u'3(x))

The following defined symbols remain to be analysed:
+'

Proved the following rewrite lemma:
+'(_gen_*':u'3(+(1, _n5)), _gen_*':u'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

Induction Base:
+'(_gen_*':u'3(+(1, 0)), _gen_*':u'3(+(1, 0)))

Induction Step:
+'(_gen_*':u'3(+(1, +(_\$n6, 1))), _gen_*':u'3(+(1, +(_\$n6, 1)))) →RΩ(1)
*'(_hole_a2, +'(_gen_*':u'3(+(1, _\$n6)), _gen_*':u'3(+(1, _\$n6)))) →IH
*'(_hole_a2, _*4)

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

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_*':u'1 :: *':u'
_hole_a2 :: a
_gen_*':u'3 :: Nat → *':u'

Lemmas:
+'(_gen_*':u'3(+(1, _n5)), _gen_*':u'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_*':u'3(0) ⇔ u'
_gen_*':u'3(+(x, 1)) ⇔ *'(_hole_a2, _gen_*':u'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
+'(_gen_*':u'3(+(1, _n5)), _gen_*':u'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)