The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(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) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

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))

(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:none
S tuples:none
K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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