(0) Obligation:

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

f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(c(f(z0)))
f(f(z0)) → f(d(f(z0)))
g(c(z0)) → z0
g(d(z0)) → z0
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))
Tuples:

F(f(z0)) → c1(F(c(f(z0))), F(z0))
F(f(z0)) → c2(F(d(f(z0))), F(z0))
G(c(z0)) → c3
G(d(z0)) → c4
G(c(0)) → c5(G(d(1)))
G(c(1)) → c6(G(d(0)))
S tuples:

F(f(z0)) → c1(F(c(f(z0))), F(z0))
F(f(z0)) → c2(F(d(f(z0))), F(z0))
G(c(z0)) → c3
G(d(z0)) → c4
G(c(0)) → c5(G(d(1)))
G(c(1)) → c6(G(d(0)))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c1, c2, c3, c4, c5, c6

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

G(c(1)) → c6(G(d(0)))
G(c(0)) → c5(G(d(1)))
G(d(z0)) → c4
G(c(z0)) → c3

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(c(f(z0)))
f(f(z0)) → f(d(f(z0)))
g(c(z0)) → z0
g(d(z0)) → z0
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))
Tuples:

F(f(z0)) → c1(F(c(f(z0))), F(z0))
F(f(z0)) → c2(F(d(f(z0))), F(z0))
S tuples:

F(f(z0)) → c1(F(c(f(z0))), F(z0))
F(f(z0)) → c2(F(d(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

(5) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(z0)) → c1(F(c(f(z0))), F(z0))
F(f(z0)) → c2(F(d(f(z0))), F(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0)) → f(c(f(z0)))
f(f(z0)) → f(d(f(z0)))
g(c(z0)) → z0
g(d(z0)) → z0
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(8) BOUNDS(1, 1)