The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).

(0) Obligation:

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

+(0, y) → y
+(s(x), y) → s(+(x, y))
+(p(x), y) → p(+(x, y))
minus(0) → 0
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*(0, y) → 0
*(s(x), y) → +(*(x, y), y)
*(p(x), y) → +(*(x, y), minus(y))

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:

+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

(5) OrderProof (LOWER BOUND(ID) transformation)

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

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

(6) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
+'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) →_R^Ω(1)
gen_0':s:p2_0(b)

Induction Step:
+'(gen_0':s:p2_0(+(n4_0, 1)), gen_0':s:p2_0(b)) →_R^Ω(1)
s(+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b))) →_IH
s(gen_0':s:p2_0(+(b, c5_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

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

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol minus.

(11) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

The following defined symbols remain to be analysed:
*'

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

Proved the following rewrite lemma:
*'(gen_0':s:p2_0(n1761_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1761_0, b)), rt ∈ Ω(1 + b·n1761₀² + n1761₀)

Induction Base:
*'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) →_R^Ω(1)
0'

Induction Step:
*'(gen_0':s:p2_0(+(n1761_0, 1)), gen_0':s:p2_0(b)) →_R^Ω(1)
+'(*'(gen_0':s:p2_0(n1761_0), gen_0':s:p2_0(b)), gen_0':s:p2_0(b)) →_IH
+'(gen_0':s:p2_0(*(c1762_0, b)), gen_0':s:p2_0(b)) →_L^{Ω(1 + b·n1761₀)}
gen_0':s:p2_0(+(*(n1761_0, b), b))

We have rt ∈ Ω(n³) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n³).

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
*'(gen_0':s:p2_0(n1761_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1761_0, b)), rt ∈ Ω(1 + b·n1761₀² + n1761₀)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
*'(gen_0':s:p2_0(n1761_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1761_0, b)), rt ∈ Ω(1 + b·n1761₀² + n1761₀)

(16) BOUNDS(n^3, INF)

(17) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
*'(gen_0':s:p2_0(n1761_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1761_0, b)), rt ∈ Ω(1 + b·n1761₀² + n1761₀)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
*'(gen_0':s:p2_0(n1761_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(*(n1761_0, b)), rt ∈ Ω(1 + b·n1761₀² + n1761₀)

(19) BOUNDS(n^3, INF)

(20) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
+'(p(x), y) → p(+'(x, y))
minus(0') → 0'
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
*'(p(x), y) → +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p → 0':s:p → 0':s:p
0' :: 0':s:p
s :: 0':s:p → 0':s:p
p :: 0':s:p → 0':s:p
minus :: 0':s:p → 0':s:p
*' :: 0':s:p → 0':s:p → 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat → 0':s:p

Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s:p2_0(0) ⇔ 0'
gen_0':s:p2_0(+(x, 1)) ⇔ s(gen_0':s:p2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) → gen_0':s:p2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(22) BOUNDS(n^1, INF)