(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(g(f(z0), z0))
f(f(z0)) → f(h(f(z0), f(z0)))
g(z0, z1) → z1
h(z0, z0) → g(z0, 0)
Tuples:
F(f(z0)) → c(F(g(f(z0), z0)), G(f(z0), z0), F(z0))
F(f(z0)) → c1(F(h(f(z0), f(z0))), H(f(z0), f(z0)), F(z0), F(z0))
H(z0, z0) → c3(G(z0, 0))
S tuples:
F(f(z0)) → c(F(g(f(z0), z0)), G(f(z0), z0), F(z0))
F(f(z0)) → c1(F(h(f(z0), f(z0))), H(f(z0), f(z0)), F(z0), F(z0))
H(z0, z0) → c3(G(z0, 0))
K tuples:none
Defined Rule Symbols:
f, g, h
Defined Pair Symbols:
F, H
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(f(z0)) → c(F(g(f(z0), z0)), G(f(z0), z0), F(z0))
F(f(z0)) → c1(F(h(f(z0), f(z0))), H(f(z0), f(z0)), F(z0), F(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(g(f(z0), z0))
f(f(z0)) → f(h(f(z0), f(z0)))
g(z0, z1) → z1
h(z0, z0) → g(z0, 0)
Tuples:
H(z0, z0) → c3(G(z0, 0))
S tuples:
H(z0, z0) → c3(G(z0, 0))
K tuples:none
Defined Rule Symbols:
f, g, h
Defined Pair Symbols:
H
Compound Symbols:
c3
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
H(z0, z0) → c3(G(z0, 0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(g(f(z0), z0))
f(f(z0)) → f(h(f(z0), f(z0)))
g(z0, z1) → z1
h(z0, z0) → g(z0, 0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, g, h
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))