(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → X
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
activate(n__f(X)) →+ f(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__f(X)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
f(f(X)) → c(n__f(n__g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
g(X) → n__g(X)
d(X) → n__d(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(X)
activate(n__d(X)) → d(X)
activate(X) → X
Types:
f :: n__f:n__g:n__d → n__f:n__g:n__d
c :: n__f:n__g:n__d → n__f:n__g:n__d
n__f :: n__f:n__g:n__d → n__f:n__g:n__d
n__g :: n__f:n__g:n__d → n__f:n__g:n__d
d :: n__f:n__g:n__d → n__f:n__g:n__d
activate :: n__f:n__g:n__d → n__f:n__g:n__d
h :: n__f:n__g:n__d → n__f:n__g:n__d
n__d :: n__f:n__g:n__d → n__f:n__g:n__d
g :: n__f:n__g:n__d → n__f:n__g:n__d
hole_n__f:n__g:n__d1_0 :: n__f:n__g:n__d
gen_n__f:n__g:n__d2_0 :: Nat → n__f:n__g:n__d
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
f,
c,
activateThey will be analysed ascendingly in the following order:
f = c
f = activate
c = activate
(8) Obligation:
TRS:
Rules:
f(
f(
X)) →
c(
n__f(
n__g(
n__f(
X))))
c(
X) →
d(
activate(
X))
h(
X) →
c(
n__d(
X))
f(
X) →
n__f(
X)
g(
X) →
n__g(
X)
d(
X) →
n__d(
X)
activate(
n__f(
X)) →
f(
activate(
X))
activate(
n__g(
X)) →
g(
X)
activate(
n__d(
X)) →
d(
X)
activate(
X) →
XTypes:
f :: n__f:n__g:n__d → n__f:n__g:n__d
c :: n__f:n__g:n__d → n__f:n__g:n__d
n__f :: n__f:n__g:n__d → n__f:n__g:n__d
n__g :: n__f:n__g:n__d → n__f:n__g:n__d
d :: n__f:n__g:n__d → n__f:n__g:n__d
activate :: n__f:n__g:n__d → n__f:n__g:n__d
h :: n__f:n__g:n__d → n__f:n__g:n__d
n__d :: n__f:n__g:n__d → n__f:n__g:n__d
g :: n__f:n__g:n__d → n__f:n__g:n__d
hole_n__f:n__g:n__d1_0 :: n__f:n__g:n__d
gen_n__f:n__g:n__d2_0 :: Nat → n__f:n__g:n__d
Generator Equations:
gen_n__f:n__g:n__d2_0(0) ⇔ hole_n__f:n__g:n__d1_0
gen_n__f:n__g:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__f:n__g:n__d2_0(x))
The following defined symbols remain to be analysed:
c, f, activate
They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol c.
(10) Obligation:
TRS:
Rules:
f(
f(
X)) →
c(
n__f(
n__g(
n__f(
X))))
c(
X) →
d(
activate(
X))
h(
X) →
c(
n__d(
X))
f(
X) →
n__f(
X)
g(
X) →
n__g(
X)
d(
X) →
n__d(
X)
activate(
n__f(
X)) →
f(
activate(
X))
activate(
n__g(
X)) →
g(
X)
activate(
n__d(
X)) →
d(
X)
activate(
X) →
XTypes:
f :: n__f:n__g:n__d → n__f:n__g:n__d
c :: n__f:n__g:n__d → n__f:n__g:n__d
n__f :: n__f:n__g:n__d → n__f:n__g:n__d
n__g :: n__f:n__g:n__d → n__f:n__g:n__d
d :: n__f:n__g:n__d → n__f:n__g:n__d
activate :: n__f:n__g:n__d → n__f:n__g:n__d
h :: n__f:n__g:n__d → n__f:n__g:n__d
n__d :: n__f:n__g:n__d → n__f:n__g:n__d
g :: n__f:n__g:n__d → n__f:n__g:n__d
hole_n__f:n__g:n__d1_0 :: n__f:n__g:n__d
gen_n__f:n__g:n__d2_0 :: Nat → n__f:n__g:n__d
Generator Equations:
gen_n__f:n__g:n__d2_0(0) ⇔ hole_n__f:n__g:n__d1_0
gen_n__f:n__g:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__f:n__g:n__d2_0(x))
The following defined symbols remain to be analysed:
activate, f
They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(12) Obligation:
TRS:
Rules:
f(
f(
X)) →
c(
n__f(
n__g(
n__f(
X))))
c(
X) →
d(
activate(
X))
h(
X) →
c(
n__d(
X))
f(
X) →
n__f(
X)
g(
X) →
n__g(
X)
d(
X) →
n__d(
X)
activate(
n__f(
X)) →
f(
activate(
X))
activate(
n__g(
X)) →
g(
X)
activate(
n__d(
X)) →
d(
X)
activate(
X) →
XTypes:
f :: n__f:n__g:n__d → n__f:n__g:n__d
c :: n__f:n__g:n__d → n__f:n__g:n__d
n__f :: n__f:n__g:n__d → n__f:n__g:n__d
n__g :: n__f:n__g:n__d → n__f:n__g:n__d
d :: n__f:n__g:n__d → n__f:n__g:n__d
activate :: n__f:n__g:n__d → n__f:n__g:n__d
h :: n__f:n__g:n__d → n__f:n__g:n__d
n__d :: n__f:n__g:n__d → n__f:n__g:n__d
g :: n__f:n__g:n__d → n__f:n__g:n__d
hole_n__f:n__g:n__d1_0 :: n__f:n__g:n__d
gen_n__f:n__g:n__d2_0 :: Nat → n__f:n__g:n__d
Generator Equations:
gen_n__f:n__g:n__d2_0(0) ⇔ hole_n__f:n__g:n__d1_0
gen_n__f:n__g:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__f:n__g:n__d2_0(x))
The following defined symbols remain to be analysed:
f
They will be analysed ascendingly in the following order:
f = c
f = activate
c = activate
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol f.
(14) Obligation:
TRS:
Rules:
f(
f(
X)) →
c(
n__f(
n__g(
n__f(
X))))
c(
X) →
d(
activate(
X))
h(
X) →
c(
n__d(
X))
f(
X) →
n__f(
X)
g(
X) →
n__g(
X)
d(
X) →
n__d(
X)
activate(
n__f(
X)) →
f(
activate(
X))
activate(
n__g(
X)) →
g(
X)
activate(
n__d(
X)) →
d(
X)
activate(
X) →
XTypes:
f :: n__f:n__g:n__d → n__f:n__g:n__d
c :: n__f:n__g:n__d → n__f:n__g:n__d
n__f :: n__f:n__g:n__d → n__f:n__g:n__d
n__g :: n__f:n__g:n__d → n__f:n__g:n__d
d :: n__f:n__g:n__d → n__f:n__g:n__d
activate :: n__f:n__g:n__d → n__f:n__g:n__d
h :: n__f:n__g:n__d → n__f:n__g:n__d
n__d :: n__f:n__g:n__d → n__f:n__g:n__d
g :: n__f:n__g:n__d → n__f:n__g:n__d
hole_n__f:n__g:n__d1_0 :: n__f:n__g:n__d
gen_n__f:n__g:n__d2_0 :: Nat → n__f:n__g:n__d
Generator Equations:
gen_n__f:n__g:n__d2_0(0) ⇔ hole_n__f:n__g:n__d1_0
gen_n__f:n__g:n__d2_0(+(x, 1)) ⇔ n__f(gen_n__f:n__g:n__d2_0(x))
No more defined symbols left to analyse.