The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳4 CdtProblem
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳6 CdtProblem
↳7 SIsEmptyProof (⇔)
↳8 BOUNDS(O(1), O(1))

(0) Obligation:

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

add(0, x) → x
add(s(x), y) → s(add(x, y))
mult(0, x) → 0
mult(s(x), y) → add(y, mult(x, y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
mult(0, z0) → 0
mult(s(z0), z1) → add(z1, mult(z0, z1))
Tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
S tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
K tuples:none
Defined Rule Symbols:

add, mult

Defined Pair Symbols:

ADD, MULT

Compound Symbols:

c1, c3

(3) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ADD(s(z0), z1) → c1(ADD(z0, z1))
We considered the (Usable) Rules:

mult(0, z0) → 0
mult(s(z0), z1) → add(z1, mult(z0, z1))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
And the Tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ADD(x1, x2)) = [2]x1   
POL(MULT(x1, x2)) = x1·x2   
POL(add(x1, x2)) = 0   
POL(c1(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   
POL(mult(x1, x2)) = 0   
POL(s(x1)) = [2] + x1   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
mult(0, z0) → 0
mult(s(z0), z1) → add(z1, mult(z0, z1))
Tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
S tuples:

MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
K tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
Defined Rule Symbols:

add, mult

Defined Pair Symbols:

ADD, MULT

Compound Symbols:

c1, c3

(5) 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.

MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
We considered the (Usable) Rules:

mult(0, z0) → 0
mult(s(z0), z1) → add(z1, mult(z0, z1))
add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
And the Tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]   
POL(ADD(x1, x2)) = 0   
POL(MULT(x1, x2)) = x1   
POL(add(x1, x2)) = [5] + [5]x1   
POL(c1(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   
POL(mult(x1, x2)) = [2] + [3]x1   
POL(s(x1)) = [2] + x1   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

add(0, z0) → z0
add(s(z0), z1) → s(add(z0, z1))
mult(0, z0) → 0
mult(s(z0), z1) → add(z1, mult(z0, z1))
Tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
S tuples:none
K tuples:

ADD(s(z0), z1) → c1(ADD(z0, z1))
MULT(s(z0), z1) → c3(ADD(z1, mult(z0, z1)), MULT(z0, z1))
Defined Rule Symbols:

add, mult

Defined Pair Symbols:

ADD, MULT

Compound Symbols:

c1, c3

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))