(0) 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
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) 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
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
P(g(g(z0))) → c1(Q(g(f(z0))))
P(f(f(z0))) → c(Q(f(g(z0))))
Q(g(g(z0))) → c3(P(g(f(z0))))
Q(f(f(z0))) → c2(P(f(g(z0))))
(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:none
S tuples:none
K tuples:none
Defined Rule Symbols:
p, q
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)