### (0) Obligation:

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

times(x, plus(y, s(z))) → plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, s(z2))) → plus(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2)))
times(z0, 0) → 0
times(z0, s(z1)) → plus(times(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, plus(z1, s(z2))) → c(PLUS(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))), TIMES(z0, plus(z1, times(s(z2), 0))), PLUS(z1, times(s(z2), 0)), TIMES(s(z2), 0), TIMES(z0, s(z2)))
TIMES(z0, 0) → c1
TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, 0) → c3
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:

TIMES(z0, plus(z1, s(z2))) → c(PLUS(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))), TIMES(z0, plus(z1, times(s(z2), 0))), PLUS(z1, times(s(z2), 0)), TIMES(s(z2), 0), TIMES(z0, s(z2)))
TIMES(z0, 0) → c1
TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, 0) → c3
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c, c1, c2, c3, c4

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

TIMES(z0, plus(z1, s(z2))) → c(PLUS(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2))), TIMES(z0, plus(z1, times(s(z2), 0))), PLUS(z1, times(s(z2), 0)), TIMES(s(z2), 0), TIMES(z0, s(z2)))
Removed 2 trailing nodes:

TIMES(z0, 0) → c1
PLUS(z0, 0) → c3

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, plus(z1, s(z2))) → plus(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2)))
times(z0, 0) → 0
times(z0, s(z1)) → plus(times(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c2, c4

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

times(z0, plus(z1, s(z2))) → plus(times(z0, plus(z1, times(s(z2), 0))), times(z0, s(z2)))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, 0) → 0
times(z0, s(z1)) → plus(times(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c2, c4

### (7) CdtRuleRemovalProof (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, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(PLUS(x1, x2)) = 0
POL(TIMES(x1, x2)) = [2]x2
POL(c2(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(plus(x1, x2)) = 0
POL(s(x1)) = [1] + x1
POL(times(x1, x2)) = [4] + [2]x1 + [2]x2

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, 0) → 0
times(z0, s(z1)) → plus(times(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:

PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
K tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c2, c4

### (9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(PLUS(x1, x2)) = x2
POL(TIMES(x1, x2)) = [2]x1·x2
POL(c2(x1, x2)) = x1 + x2
POL(c4(x1)) = x1
POL(plus(x1, x2)) = 0
POL(s(x1)) = [1] + x1
POL(times(x1, x2)) = 0

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

times(z0, 0) → 0
times(z0, s(z1)) → plus(times(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
S tuples:none
K tuples:

TIMES(z0, s(z1)) → c2(PLUS(times(z0, z1), z0), TIMES(z0, z1))
PLUS(z0, s(z1)) → c4(PLUS(z0, z1))
Defined Rule Symbols:

times, plus

Defined Pair Symbols:

TIMES, PLUS

Compound Symbols:

c2, c4

### (11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty