(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
b → c
g(X) → n__g(X)
activate(n__g(X)) → g(activate(X))
activate(X) → X
Rewrite Strategy: INNERMOST
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
activate(n__g(X)) →+ g(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__g(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(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
b → c
g(X) → n__g(X)
activate(n__g(X)) → g(activate(X))
activate(X) → X
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
f(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
b → c
g(X) → n__g(X)
activate(n__g(X)) → g(activate(X))
activate(X) → X
Types:
f :: n__g:c → n__g:c → n__g:c → f
n__g :: n__g:c → n__g:c
activate :: n__g:c → n__g:c
g :: n__g:c → n__g:c
b :: n__g:c
c :: n__g:c
hole_f1_0 :: f
hole_n__g:c2_0 :: n__g:c
gen_n__g:c3_0 :: Nat → n__g:c
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
f,
activateThey will be analysed ascendingly in the following order:
activate < f
(8) Obligation:
Innermost TRS:
Rules:
f(
X,
n__g(
X),
Y) →
f(
activate(
Y),
activate(
Y),
activate(
Y))
g(
b) →
cb →
cg(
X) →
n__g(
X)
activate(
n__g(
X)) →
g(
activate(
X))
activate(
X) →
XTypes:
f :: n__g:c → n__g:c → n__g:c → f
n__g :: n__g:c → n__g:c
activate :: n__g:c → n__g:c
g :: n__g:c → n__g:c
b :: n__g:c
c :: n__g:c
hole_f1_0 :: f
hole_n__g:c2_0 :: n__g:c
gen_n__g:c3_0 :: Nat → n__g:c
Generator Equations:
gen_n__g:c3_0(0) ⇔ c
gen_n__g:c3_0(+(x, 1)) ⇔ n__g(gen_n__g:c3_0(x))
The following defined symbols remain to be analysed:
activate, f
They will be analysed ascendingly in the following order:
activate < f
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(10) Obligation:
Innermost TRS:
Rules:
f(
X,
n__g(
X),
Y) →
f(
activate(
Y),
activate(
Y),
activate(
Y))
g(
b) →
cb →
cg(
X) →
n__g(
X)
activate(
n__g(
X)) →
g(
activate(
X))
activate(
X) →
XTypes:
f :: n__g:c → n__g:c → n__g:c → f
n__g :: n__g:c → n__g:c
activate :: n__g:c → n__g:c
g :: n__g:c → n__g:c
b :: n__g:c
c :: n__g:c
hole_f1_0 :: f
hole_n__g:c2_0 :: n__g:c
gen_n__g:c3_0 :: Nat → n__g:c
Generator Equations:
gen_n__g:c3_0(0) ⇔ c
gen_n__g:c3_0(+(x, 1)) ⇔ n__g(gen_n__g:c3_0(x))
The following defined symbols remain to be analysed:
f
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol f.
(12) Obligation:
Innermost TRS:
Rules:
f(
X,
n__g(
X),
Y) →
f(
activate(
Y),
activate(
Y),
activate(
Y))
g(
b) →
cb →
cg(
X) →
n__g(
X)
activate(
n__g(
X)) →
g(
activate(
X))
activate(
X) →
XTypes:
f :: n__g:c → n__g:c → n__g:c → f
n__g :: n__g:c → n__g:c
activate :: n__g:c → n__g:c
g :: n__g:c → n__g:c
b :: n__g:c
c :: n__g:c
hole_f1_0 :: f
hole_n__g:c2_0 :: n__g:c
gen_n__g:c3_0 :: Nat → n__g:c
Generator Equations:
gen_n__g:c3_0(0) ⇔ c
gen_n__g:c3_0(+(x, 1)) ⇔ n__g(gen_n__g:c3_0(x))
No more defined symbols left to analyse.