The Runtime Complexity (full) 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, z)) → +(*(x, y), *(x, z))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
*' :: a → +' → +'
+' :: +' → +' → +'
hole_+'1_0 :: +'
hole_a2_0 :: a
gen_+'3_0 :: Nat → +'

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

Heuristically decided to analyse the following defined symbols:
*'

(8) Obligation:

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

Types:
*' :: a → +' → +'
+' :: +' → +' → +'
hole_+'1_0 :: +'
hole_a2_0 :: a
gen_+'3_0 :: Nat → +'

Generator Equations:
gen_+'3_0(0) ⇔ hole_+'1_0
gen_+'3_0(+(x, 1)) ⇔ +'(hole_+'1_0, gen_+'3_0(x))

The following defined symbols remain to be analysed:
*'

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol *'.

(10) Obligation:

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

Types:
*' :: a → +' → +'
+' :: +' → +' → +'
hole_+'1_0 :: +'
hole_a2_0 :: a
gen_+'3_0 :: Nat → +'

Generator Equations:
gen_+'3_0(0) ⇔ hole_+'1_0
gen_+'3_0(+(x, 1)) ⇔ +'(hole_+'1_0, gen_+'3_0(x))

No more defined symbols left to analyse.