### (0) Obligation:

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

g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
F(f(z0)) → c3(F(d(f(z0))), F(z0))
F(z0) → c4
S tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
F(f(z0)) → c3(F(d(f(z0))), F(z0))
F(z0) → c4
K tuples:none
Defined Rule Symbols:

g, f

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2, c3, c4

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

F(f(z0)) → c3(F(d(f(z0))), F(z0))
Removed 1 trailing nodes:

F(z0) → c4

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

g, f

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

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

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

### (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(s(z0), z1)) → c2(F(c(z0, s(z1))))
We considered the (Usable) Rules:none
And the Tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1)) = [4]x1
POL(G(x1)) = 0
POL(c(x1, x2)) = x1
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(s(x1)) = [2] + x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
K tuples:

F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

### (9) 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.

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

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1)) = 0
POL(G(x1)) = [4]x1
POL(c(x1, x2)) = x2
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(s(x1)) = [4] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

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

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

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

The set S is empty