(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:
f(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0), z0) → f(z0, a(z0))
f(z0, c(z0)) → f(s(z0), z0)
f(z0, z0) → c(z0)
Tuples:
F(s(z0), z0) → c1(F(z0, a(z0)))
F(z0, c(z0)) → c2(F(s(z0), z0))
F(z0, z0) → c3
S tuples:
F(s(z0), z0) → c1(F(z0, a(z0)))
F(z0, c(z0)) → c2(F(s(z0), z0))
F(z0, z0) → c3
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, c2, c3
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
F(z0, z0) → c3
F(s(z0), z0) → c1(F(z0, a(z0)))
F(z0, c(z0)) → c2(F(s(z0), z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0), z0) → f(z0, a(z0))
f(z0, c(z0)) → f(s(z0), z0)
f(z0, z0) → c(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)