The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +(S(0), add0(x', x))
power(x, 0) → S(0)
mult(x, 0) → 0
add0(x, 0) → x

The (relative) TRS S consists of the following rules:

+(x, S(0)) → S(x)
+(S(0), y) → S(y)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
power(x', S(x)) →⁺ mult(x', power(x', x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / S(x)].
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:

power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x

The (relative) TRS S consists of the following rules:

+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
power, mult, add0

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

(8) Obligation:

Innermost TRS:
Rules:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

The following defined symbols remain to be analysed:
add0, power, mult

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

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

Proved the following rewrite lemma:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)

Induction Base:
add0(gen_S:0'2_1(1), gen_S:0'2_1(0)) →_R^Ω(1)
gen_S:0'2_1(1)

Induction Step:
add0(gen_S:0'2_1(1), gen_S:0'2_1(+(n4_1, 1))) →_R^Ω(1)
+'(S(0'), add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1))) →_IH
+'(S(0'), gen_S:0'2_1(+(1, c5_1))) →_R^Ω(0)
S(gen_S:0'2_1(+(1, n4_1)))

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:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

The following defined symbols remain to be analysed:
mult, power

They will be analysed ascendingly in the following order:
mult < power

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)

Induction Base:
mult(gen_S:0'2_1(1), gen_S:0'2_1(0)) →_R^Ω(1)
0'

Induction Step:
mult(gen_S:0'2_1(1), gen_S:0'2_1(+(n487_1, 1))) →_R^Ω(1)
add0(gen_S:0'2_1(1), mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1))) →_IH
add0(gen_S:0'2_1(1), gen_S:0'2_1(c488_1)) →_L^{Ω(1 + n487₁)}
gen_S:0'2_1(+(1, n487_1))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

The following defined symbols remain to be analysed:
power

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
power(gen_S:0'2_1(1), gen_S:0'2_1(n780_1)) → gen_S:0'2_1(1), rt ∈ Ω(1 + n780₁)

Induction Base:
power(gen_S:0'2_1(1), gen_S:0'2_1(0)) →_R^Ω(1)
S(0')

Induction Step:
power(gen_S:0'2_1(1), gen_S:0'2_1(+(n780_1, 1))) →_R^Ω(1)
mult(gen_S:0'2_1(1), power(gen_S:0'2_1(1), gen_S:0'2_1(n780_1))) →_IH
mult(gen_S:0'2_1(1), gen_S:0'2_1(1)) →_L^Ω(3)
gen_S:0'2_1(1)

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)
power(gen_S:0'2_1(1), gen_S:0'2_1(n780_1)) → gen_S:0'2_1(1), rt ∈ Ω(1 + n780₁)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)

(19) BOUNDS(n^2, INF)

(20) Obligation:

Innermost TRS:
Rules:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)
power(gen_S:0'2_1(1), gen_S:0'2_1(n780_1)) → gen_S:0'2_1(1), rt ∈ Ω(1 + n780₁)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost TRS:
Rules:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) → gen_S:0'2_1(n487_1), rt ∈ Ω(1 + n487₁ + n487₁²)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Innermost TRS:
Rules:
power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +'(S(0'), add0(x', x))
power(x, 0') → S(0')
mult(x, 0') → 0'
add0(x, 0') → x
+'(x, S(0')) → S(x)
+'(S(0'), y) → S(y)

Types:
power :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
mult :: S:0' → S:0' → S:0'
add0 :: S:0' → S:0' → S:0'
+' :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(+(1, n4_1)), rt ∈ Ω(1 + n4₁)

(28) BOUNDS(n^1, INF)