(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a, h(x)) → f(g(x), h(x))
h(g(x)) → h(a)
g(h(x)) → g(x)
h(h(x)) → x
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, h(z0)) → f(g(z0), h(z0))
h(g(z0)) → h(a)
h(h(z0)) → z0
g(h(z0)) → g(z0)
Tuples:
F(a, h(z0)) → c(F(g(z0), h(z0)), G(z0), H(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))
S tuples:
F(a, h(z0)) → c(F(g(z0), h(z0)), G(z0), H(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))
K tuples:none
Defined Rule Symbols:
f, h, g
Defined Pair Symbols:
F, H, G
Compound Symbols:
c, c1, c3
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(a, h(z0)) → c(F(g(z0), h(z0)), G(z0), H(z0))
H(g(z0)) → c1(H(a))
G(h(z0)) → c3(G(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, h(z0)) → f(g(z0), h(z0))
h(g(z0)) → h(a)
h(h(z0)) → z0
g(h(z0)) → g(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, h, g
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))