(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(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(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
F(a(g(z0))) → c1(B(z0))
B(z0) → c2
S tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
F(a(g(z0))) → c1(B(z0))
B(z0) → c2
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F, B
Compound Symbols:
c, c1, c2
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(a(g(z0))) → c1(B(z0))
B(z0) → c2
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
S tuples:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(f(z0)) → c(F(a(b(f(z0)))), B(f(z0)), F(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(a(b(f(z0))))
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)