### (0) Obligation:

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

f(c(X, s(Y))) → f(c(s(X), Y))
g(c(s(X), Y)) → f(c(X, s(Y)))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, s(z1))) → f(c(s(z0), z1))
g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
S tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c1, c2

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

G(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, s(z1))) → f(c(s(z0), z1))
g(c(s(z0), z1)) → f(c(z0, s(z1)))
Tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
S tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c1

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(c(z0, s(z1))) → f(c(s(z0), z1))
g(c(s(z0), z1)) → f(c(z0, s(z1)))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
S tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c1

### (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.

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
We considered the (Usable) Rules:none
And the Tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1)) = [4]x1
POL(c(x1, x2)) = x2
POL(c1(x1)) = x1
POL(s(x1)) = [4] + x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
S tuples:none
K tuples:

F(c(z0, s(z1))) → c1(F(c(s(z0), z1)))
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c1

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

The set S is empty