### (0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).

The TRS R consists of the following rules:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

### (2) Obligation:

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

The TRS R consists of the following rules:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(f(f(z0))) → q(f(g(z0)))
p(g(g(z0))) → q(g(f(z0)))
q(f(f(z0))) → p(f(g(z0)))
q(g(g(z0))) → p(g(f(z0)))
Tuples:

P(f(f(z0))) → c(Q(f(g(z0))))
P(g(g(z0))) → c1(Q(g(f(z0))))
Q(f(f(z0))) → c2(P(f(g(z0))))
Q(g(g(z0))) → c3(P(g(f(z0))))
S tuples:

P(f(f(z0))) → c(Q(f(g(z0))))
P(g(g(z0))) → c1(Q(g(f(z0))))
Q(f(f(z0))) → c2(P(f(g(z0))))
Q(g(g(z0))) → c3(P(g(f(z0))))
K tuples:none
Defined Rule Symbols:

p, q

Defined Pair Symbols:

P, Q

Compound Symbols:

c, c1, c2, c3

### (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

Q(g(g(z0))) → c3(P(g(f(z0))))
P(f(f(z0))) → c(Q(f(g(z0))))
P(g(g(z0))) → c1(Q(g(f(z0))))
Q(f(f(z0))) → c2(P(f(g(z0))))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(f(f(z0))) → q(f(g(z0)))
p(g(g(z0))) → q(g(f(z0)))
q(f(f(z0))) → p(f(g(z0)))
q(g(g(z0))) → p(g(f(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

p, q

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty