The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

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

p(s(x)) → x
fac(0) → s(0)
fac(s(x)) → times(s(x), fac(p(s(x))))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))

Tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))

S tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))

K tuples:none
Defined Rule Symbols:

p, fac

Defined Pair Symbols:

FAC

Compound Symbols:

c2

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

Use narrowing to replace FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0))) by

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))
FAC(s(x0)) → c2

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))

Tuples:

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))
FAC(s(x0)) → c2

S tuples:

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))
FAC(s(x0)) → c2

K tuples:none
Defined Rule Symbols:

p, fac

Defined Pair Symbols:

FAC

Compound Symbols:

c2, c2

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

Removed 1 trailing nodes:

FAC(s(x0)) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))

Tuples:

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))

S tuples:

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))

K tuples:none
Defined Rule Symbols:

p, fac

Defined Pair Symbols:

FAC

Compound Symbols:

c2

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))

We considered the (Usable) Rules:none
And the Tuples:

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(FAC(x₁)) = [4]x₁   
POL(P(x₁)) = 0   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(s(x₁)) = [4] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))

Tuples:

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))

S tuples:none
K tuples:

FAC(s(z0)) → c2(FAC(z0), P(s(z0)))

Defined Rule Symbols:

p, fac

Defined Pair Symbols:

FAC

Compound Symbols:

c2

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))