(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(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) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 0th argument of d: d, f, activate
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(f(X)) → c(n__f(g(n__f(X))))
(2) 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:
activate(n__d(X)) → d(X)
activate(n__f(X)) → f(X)
f(X) → n__f(X)
d(X) → n__d(X)
h(X) → c(n__d(X))
activate(X) → X
c(X) → d(activate(X))
Rewrite Strategy: INNERMOST
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__d(z0)) → d(z0)
activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
d(z0) → n__d(z0)
h(z0) → c(n__d(z0))
c(z0) → d(activate(z0))
Tuples:
ACTIVATE(n__d(z0)) → c1(D(z0))
ACTIVATE(n__f(z0)) → c2(F(z0))
ACTIVATE(z0) → c3
F(z0) → c4
D(z0) → c5
H(z0) → c6(C(n__d(z0)))
C(z0) → c7(D(activate(z0)), ACTIVATE(z0))
S tuples:
ACTIVATE(n__d(z0)) → c1(D(z0))
ACTIVATE(n__f(z0)) → c2(F(z0))
ACTIVATE(z0) → c3
F(z0) → c4
D(z0) → c5
H(z0) → c6(C(n__d(z0)))
C(z0) → c7(D(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
activate, f, d, h, c
Defined Pair Symbols:
ACTIVATE, F, D, H, C
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing nodes:
D(z0) → c5
ACTIVATE(z0) → c3
ACTIVATE(n__d(z0)) → c1(D(z0))
C(z0) → c7(D(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c2(F(z0))
F(z0) → c4
H(z0) → c6(C(n__d(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__d(z0)) → d(z0)
activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
d(z0) → n__d(z0)
h(z0) → c(n__d(z0))
c(z0) → d(activate(z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
activate, f, d, h, c
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)