(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(z0) → n__f(z0)
a → n__a
g(z0) → n__g(z0)
activate(n__f(z0)) → f(z0)
activate(n__a) → a
activate(n__g(z0)) → g(z0)
activate(z0) → z0
Tuples:
F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__g(z0)) → c6(G(z0))
S tuples:
F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__g(z0)) → c6(G(z0))
K tuples:none
Defined Rule Symbols:
f, a, g, activate
Defined Pair Symbols:
F, ACTIVATE
Compound Symbols:
c, c4, c5, c6
(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 4 of 4 dangling nodes:
F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__g(z0)) → c6(G(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(z0) → n__f(z0)
a → n__a
g(z0) → n__g(z0)
activate(n__f(z0)) → f(z0)
activate(n__a) → a
activate(n__g(z0)) → g(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, a, g, activate
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))