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:

times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, 1)) → plus(times(z0, plus(z1, times(1, 0))), z0)
times(z0, 1) → z0
times(z0, 0) → 0
plus(z0, 0) → z0

Tuples:

TIMES(z0, plus(z1, 1)) → c(PLUS(times(z0, plus(z1, times(1, 0))), z0), TIMES(z0, plus(z1, times(1, 0))), PLUS(z1, times(1, 0)), TIMES(1, 0))

S tuples:

TIMES(z0, plus(z1, 1)) → c(PLUS(times(z0, plus(z1, times(1, 0))), z0), TIMES(z0, plus(z1, times(1, 0))), PLUS(z1, times(1, 0)), TIMES(1, 0))

K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES

Compound Symbols:

c

(3) 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.

TIMES(z0, plus(z1, 1)) → c(PLUS(times(z0, plus(z1, times(1, 0))), z0), TIMES(z0, plus(z1, times(1, 0))), PLUS(z1, times(1, 0)), TIMES(1, 0))

We considered the (Usable) Rules:

times(z0, 0) → 0
times(z0, plus(z1, 1)) → plus(times(z0, plus(z1, times(1, 0))), z0)
times(z0, 1) → z0
plus(z0, 0) → z0

And the Tuples:

TIMES(z0, plus(z1, 1)) → c(PLUS(times(z0, plus(z1, times(1, 0))), z0), TIMES(z0, plus(z1, times(1, 0))), PLUS(z1, times(1, 0)), TIMES(1, 0))

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

POL(0) = 0   
POL(1) = [4]   
POL(PLUS(x₁, x₂)) = 0   
POL(TIMES(x₁, x₂)) = [4]x₂   
POL(c(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(plus(x₁, x₂)) = [3] + [2]x₁ + [2]x₂   
POL(times(x₁, x₂)) = x₂   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, 1)) → plus(times(z0, plus(z1, times(1, 0))), z0)
times(z0, 1) → z0
times(z0, 0) → 0
plus(z0, 0) → z0

Tuples:

TIMES(z0, plus(z1, 1)) → c(PLUS(times(z0, plus(z1, times(1, 0))), z0), TIMES(z0, plus(z1, times(1, 0))), PLUS(z1, times(1, 0)), TIMES(1, 0))

S tuples:none
K tuples:

TIMES(z0, plus(z1, 1)) → c(PLUS(times(z0, plus(z1, times(1, 0))), z0), TIMES(z0, plus(z1, times(1, 0))), PLUS(z1, times(1, 0)), TIMES(1, 0))

Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES

Compound Symbols:

c

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))