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, z)) → *(otimes(x, y), z)
*(1, y) → y
*(+(x, y), z) → oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) → oplus(*(x, y), *(x, z))

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, z)) → *'(otimes(x, y), z)
*'(1', y) → y
*'(+'(x, y), z) → oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) → oplus(*'(x, y), *'(x, z))

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
otimes/0
otimes/1

(4) Obligation:

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

*'(x, *'(y, z)) → *'(otimes, z)
*'(1', y) → y
*'(+'(x, y), z) → oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) → oplus(*'(x, y), *'(x, z))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
*'(x, *'(y, z)) → *'(otimes, z)
*'(1', y) → y
*'(+'(x, y), z) → oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) → oplus(*'(x, y), *'(x, z))

Types:
*' :: otimes:1':+' → oplus → oplus
otimes :: otimes:1':+'
1' :: otimes:1':+'
+' :: otimes:1':+' → otimes:1':+' → otimes:1':+'
oplus :: oplus → oplus → oplus
hole_oplus1_0 :: oplus
hole_otimes:1':+'2_0 :: otimes:1':+'
gen_oplus3_0 :: Nat → oplus
gen_otimes:1':+'4_0 :: Nat → otimes:1':+'

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

Heuristically decided to analyse the following defined symbols:
*'

(8) Obligation:

Innermost TRS:
Rules:
*'(x, *'(y, z)) → *'(otimes, z)
*'(1', y) → y
*'(+'(x, y), z) → oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) → oplus(*'(x, y), *'(x, z))

Types:
*' :: otimes:1':+' → oplus → oplus
otimes :: otimes:1':+'
1' :: otimes:1':+'
+' :: otimes:1':+' → otimes:1':+' → otimes:1':+'
oplus :: oplus → oplus → oplus
hole_oplus1_0 :: oplus
hole_otimes:1':+'2_0 :: otimes:1':+'
gen_oplus3_0 :: Nat → oplus
gen_otimes:1':+'4_0 :: Nat → otimes:1':+'

Generator Equations:
gen_oplus3_0(0) ⇔ hole_oplus1_0
gen_oplus3_0(+(x, 1)) ⇔ oplus(hole_oplus1_0, gen_oplus3_0(x))
gen_otimes:1':+'4_0(0) ⇔ 1'
gen_otimes:1':+'4_0(+(x, 1)) ⇔ +'(1', gen_otimes:1':+'4_0(x))

The following defined symbols remain to be analysed:
*'

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

Induction Base:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(0))

Induction Step:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(+(n6_0, 1))) →_R^Ω(1)
oplus(*'(gen_otimes:1':+'4_0(0), hole_oplus1_0), *'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0))) →_R^Ω(1)
oplus(hole_oplus1_0, *'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0))) →_IH
oplus(hole_oplus1_0, *5_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
*'(x, *'(y, z)) → *'(otimes, z)
*'(1', y) → y
*'(+'(x, y), z) → oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) → oplus(*'(x, y), *'(x, z))

Types:
*' :: otimes:1':+' → oplus → oplus
otimes :: otimes:1':+'
1' :: otimes:1':+'
+' :: otimes:1':+' → otimes:1':+' → otimes:1':+'
oplus :: oplus → oplus → oplus
hole_oplus1_0 :: oplus
hole_otimes:1':+'2_0 :: otimes:1':+'
gen_oplus3_0 :: Nat → oplus
gen_otimes:1':+'4_0 :: Nat → otimes:1':+'

Lemmas:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_oplus3_0(0) ⇔ hole_oplus1_0
gen_oplus3_0(+(x, 1)) ⇔ oplus(hole_oplus1_0, gen_oplus3_0(x))
gen_otimes:1':+'4_0(0) ⇔ 1'
gen_otimes:1':+'4_0(+(x, 1)) ⇔ +'(1', gen_otimes:1':+'4_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
*'(x, *'(y, z)) → *'(otimes, z)
*'(1', y) → y
*'(+'(x, y), z) → oplus(*'(x, z), *'(y, z))
*'(x, oplus(y, z)) → oplus(*'(x, y), *'(x, z))

Types:
*' :: otimes:1':+' → oplus → oplus
otimes :: otimes:1':+'
1' :: otimes:1':+'
+' :: otimes:1':+' → otimes:1':+' → otimes:1':+'
oplus :: oplus → oplus → oplus
hole_oplus1_0 :: oplus
hole_otimes:1':+'2_0 :: otimes:1':+'
gen_oplus3_0 :: Nat → oplus
gen_otimes:1':+'4_0 :: Nat → otimes:1':+'

Lemmas:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

Generator Equations:
gen_oplus3_0(0) ⇔ hole_oplus1_0
gen_oplus3_0(+(x, 1)) ⇔ oplus(hole_oplus1_0, gen_oplus3_0(x))
gen_otimes:1':+'4_0(0) ⇔ 1'
gen_otimes:1':+'4_0(+(x, 1)) ⇔ +'(1', gen_otimes:1':+'4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_otimes:1':+'4_0(0), gen_oplus3_0(n6_0)) → *5_0, rt ∈ Ω(n6₀)

(16) BOUNDS(n^1, INF)