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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
+(*(x3385_0, y), *(x3385_0, z)) →⁺ *(x3385_0, +(y, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [y / *(x3385_0, y), z / *(x3385_0, z)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)