(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0, 1, g(x, y), z) → f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) → 0
g(0, 1) → 1
h(g(x, y)) → h(x)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, g(z0, z1), z2) → f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
g(0, 1) → 0
g(0, 1) → 1
h(g(z0, z1)) → h(z0)
Tuples:
F(0, 1, g(z0, z1), z2) → c(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
H(g(z0, z1)) → c3(H(z0))
S tuples:
F(0, 1, g(z0, z1), z2) → c(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
H(g(z0, z1)) → c3(H(z0))
K tuples:none
Defined Rule Symbols:
f, g, h
Defined Pair Symbols:
F, H
Compound Symbols:
c, c3
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(0, 1, g(z0, z1), z2) → c(F(g(z0, z1), g(z0, z1), g(z0, z1), h(z0)), G(z0, z1), G(z0, z1), G(z0, z1), H(z0))
H(g(z0, z1)) → c3(H(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, g(z0, z1), z2) → f(g(z0, z1), g(z0, z1), g(z0, z1), h(z0))
g(0, 1) → 0
g(0, 1) → 1
h(g(z0, z1)) → h(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, g, h
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))