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) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
*'/0

(6) Obligation:

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

*'(+'(y, z)) → +'(*'(y), *'(z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

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

Types:
*' :: +' → +'
+' :: +' → +' → +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat → +'

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
*'

(10) Obligation:

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

Types:
*' :: +' → +'
+' :: +' → +' → +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat → +'

Generator Equations:
gen_+'2_0(0) ⇔ hole_+'1_0
gen_+'2_0(+(x, 1)) ⇔ +'(hole_+'1_0, gen_+'2_0(x))

The following defined symbols remain to be analysed:
*'

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
*'(gen_+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
*'(gen_+'2_0(+(1, 0)))

Induction Step:
*'(gen_+'2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
+'(*'(hole_+'1_0), *'(gen_+'2_0(+(1, n4_0)))) →_IH
+'(*'(hole_+'1_0), *3_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
*' :: +' → +'
+' :: +' → +' → +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat → +'

Lemmas:
*'(gen_+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_+'2_0(0) ⇔ hole_+'1_0
gen_+'2_0(+(x, 1)) ⇔ +'(hole_+'1_0, gen_+'2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
*' :: +' → +'
+' :: +' → +' → +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat → +'

Lemmas:
*'(gen_+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_+'2_0(0) ⇔ hole_+'1_0
gen_+'2_0(+(x, 1)) ⇔ +'(hole_+'1_0, gen_+'2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_+'2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(18) BOUNDS(n^1, INF)