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:

O(0) → 0
+(0, x) → x
+(x, 0) → x
+(O(x), O(y)) → O(+(x, y))
+(O(x), I(y)) → I(+(x, y))
+(I(x), O(y)) → I(+(x, y))
+(I(x), I(y)) → O(+(+(x, y), I(0)))
*(0, x) → 0
*(x, 0) → 0
*(O(x), y) → O(*(x, y))
*(I(x), y) → +(O(*(x, y)), y)
-(x, 0) → x
-(0, x) → 0
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1)))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Types:
O :: 0':I:1' → 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' → 0':I:1' → 0':I:1'
I :: 0':I:1' → 0':I:1'
*' :: 0':I:1' → 0':I:1' → 0':I:1'
- :: 0':I:1' → 0':I:1' → 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat → 0':I:1'

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

Heuristically decided to analyse the following defined symbols:
+', *', -

They will be analysed ascendingly in the following order:
+' < *'

(8) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Types:
O :: 0':I:1' → 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' → 0':I:1' → 0':I:1'
I :: 0':I:1' → 0':I:1'
*' :: 0':I:1' → 0':I:1' → 0':I:1'
- :: 0':I:1' → 0':I:1' → 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat → 0':I:1'

Generator Equations:
gen_0':I:1'2_0(0) ⇔ 0'
gen_0':I:1'2_0(+(x, 1)) ⇔ I(gen_0':I:1'2_0(x))

The following defined symbols remain to be analysed:
+', *', -

They will be analysed ascendingly in the following order:
+' < *'

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

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

(10) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Types:
O :: 0':I:1' → 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' → 0':I:1' → 0':I:1'
I :: 0':I:1' → 0':I:1'
*' :: 0':I:1' → 0':I:1' → 0':I:1'
- :: 0':I:1' → 0':I:1' → 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat → 0':I:1'

Generator Equations:
gen_0':I:1'2_0(0) ⇔ 0'
gen_0':I:1'2_0(+(x, 1)) ⇔ I(gen_0':I:1'2_0(x))

The following defined symbols remain to be analysed:
*', -

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

Proved the following rewrite lemma:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)

Induction Base:
*'(gen_0':I:1'2_0(0), gen_0':I:1'2_0(0)) →_R^Ω(1)
0'

Induction Step:
*'(gen_0':I:1'2_0(+(n41051_0, 1)), gen_0':I:1'2_0(0)) →_R^Ω(1)
+'(O(*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0))), gen_0':I:1'2_0(0)) →_IH
+'(O(gen_0':I:1'2_0(0)), gen_0':I:1'2_0(0)) →_R^Ω(1)
+'(0', gen_0':I:1'2_0(0)) →_R^Ω(1)
gen_0':I:1'2_0(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:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Types:
O :: 0':I:1' → 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' → 0':I:1' → 0':I:1'
I :: 0':I:1' → 0':I:1'
*' :: 0':I:1' → 0':I:1' → 0':I:1'
- :: 0':I:1' → 0':I:1' → 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat → 0':I:1'

Lemmas:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)

Generator Equations:
gen_0':I:1'2_0(0) ⇔ 0'
gen_0':I:1'2_0(+(x, 1)) ⇔ I(gen_0':I:1'2_0(x))

The following defined symbols remain to be analysed:
-

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
-(gen_0':I:1'2_0(n46412_0), gen_0':I:1'2_0(n46412_0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n46412₀)

Induction Base:
-(gen_0':I:1'2_0(0), gen_0':I:1'2_0(0)) →_R^Ω(1)
gen_0':I:1'2_0(0)

Induction Step:
-(gen_0':I:1'2_0(+(n46412_0, 1)), gen_0':I:1'2_0(+(n46412_0, 1))) →_R^Ω(1)
O(-(gen_0':I:1'2_0(n46412_0), gen_0':I:1'2_0(n46412_0))) →_IH
O(gen_0':I:1'2_0(0)) →_R^Ω(1)
0'

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

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Types:
O :: 0':I:1' → 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' → 0':I:1' → 0':I:1'
I :: 0':I:1' → 0':I:1'
*' :: 0':I:1' → 0':I:1' → 0':I:1'
- :: 0':I:1' → 0':I:1' → 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat → 0':I:1'

Lemmas:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)
-(gen_0':I:1'2_0(n46412_0), gen_0':I:1'2_0(n46412_0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n46412₀)

Generator Equations:
gen_0':I:1'2_0(0) ⇔ 0'
gen_0':I:1'2_0(+(x, 1)) ⇔ I(gen_0':I:1'2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Types:
O :: 0':I:1' → 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' → 0':I:1' → 0':I:1'
I :: 0':I:1' → 0':I:1'
*' :: 0':I:1' → 0':I:1' → 0':I:1'
- :: 0':I:1' → 0':I:1' → 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat → 0':I:1'

Lemmas:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)
-(gen_0':I:1'2_0(n46412_0), gen_0':I:1'2_0(n46412_0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n46412₀)

Generator Equations:
gen_0':I:1'2_0(0) ⇔ 0'
gen_0':I:1'2_0(+(x, 1)) ⇔ I(gen_0':I:1'2_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
O(0') → 0'
+'(0', x) → x
+'(x, 0') → x
+'(O(x), O(y)) → O(+'(x, y))
+'(O(x), I(y)) → I(+'(x, y))
+'(I(x), O(y)) → I(+'(x, y))
+'(I(x), I(y)) → O(+'(+'(x, y), I(0')))
*'(0', x) → 0'
*'(x, 0') → 0'
*'(O(x), y) → O(*'(x, y))
*'(I(x), y) → +'(O(*'(x, y)), y)
-(x, 0') → x
-(0', x) → 0'
-(O(x), O(y)) → O(-(x, y))
-(O(x), I(y)) → I(-(-(x, y), I(1')))
-(I(x), O(y)) → I(-(x, y))
-(I(x), I(y)) → O(-(x, y))

Types:
O :: 0':I:1' → 0':I:1'
0' :: 0':I:1'
+' :: 0':I:1' → 0':I:1' → 0':I:1'
I :: 0':I:1' → 0':I:1'
*' :: 0':I:1' → 0':I:1' → 0':I:1'
- :: 0':I:1' → 0':I:1' → 0':I:1'
1' :: 0':I:1'
hole_0':I:1'1_0 :: 0':I:1'
gen_0':I:1'2_0 :: Nat → 0':I:1'

Lemmas:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)

Generator Equations:
gen_0':I:1'2_0(0) ⇔ 0'
gen_0':I:1'2_0(+(x, 1)) ⇔ I(gen_0':I:1'2_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
*'(gen_0':I:1'2_0(n41051_0), gen_0':I:1'2_0(0)) → gen_0':I:1'2_0(0), rt ∈ Ω(1 + n41051₀)

(24) BOUNDS(n^1, INF)