(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(X)) → c(n__f(g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
d(X) → n__d(X)
activate(n__f(X)) → f(X)
activate(n__d(X)) → d(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(f(z0)) → c(n__f(g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
d(z0) → n__d(z0)
activate(n__f(z0)) → f(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:
F(f(z0)) → c1(C(n__f(g(n__f(z0)))))
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
S tuples:
F(f(z0)) → c1(C(n__f(g(n__f(z0)))))
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
K tuples:none
Defined Rule Symbols:
f, c, h, d, activate
Defined Pair Symbols:
F, C, H, ACTIVATE
Compound Symbols:
c1, c3, c4, c6, c7
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(f(z0)) → c1(C(n__f(g(n__f(z0)))))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → c(n__f(g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
d(z0) → n__d(z0)
activate(n__f(z0)) → f(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
S tuples:
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
K tuples:none
Defined Rule Symbols:
f, c, h, d, activate
Defined Pair Symbols:
C, H, ACTIVATE
Compound Symbols:
c3, c4, c6, c7
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 4 of 4 dangling nodes:
C(z0) → c3(D(activate(z0)), ACTIVATE(z0))
H(z0) → c4(C(n__d(z0)))
ACTIVATE(n__f(z0)) → c6(F(z0))
ACTIVATE(n__d(z0)) → c7(D(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → c(n__f(g(n__f(z0))))
f(z0) → n__f(z0)
c(z0) → d(activate(z0))
h(z0) → c(n__d(z0))
d(z0) → n__d(z0)
activate(n__f(z0)) → f(z0)
activate(n__d(z0)) → d(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, c, h, d, activate
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))