(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(f(a))) → mark(f(g(f(a))))
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
g(mark(X)) →+ mark(g(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(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:
active(f(f(a))) → mark(f(g(f(a))))
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
active(f(f(a))) → mark(f(g(f(a))))
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X)) → f(proper(X))
proper(a) → ok(a)
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: a:mark:ok → a:mark:ok
f :: a:mark:ok → a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok → a:mark:ok
g :: a:mark:ok → a:mark:ok
proper :: a:mark:ok → a:mark:ok
ok :: a:mark:ok → a:mark:ok
top :: a:mark:ok → top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat → a:mark:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
f,
g,
proper,
topThey will be analysed ascendingly in the following order:
f < active
g < active
active < top
f < proper
g < proper
proper < top
(8) Obligation:
Innermost TRS:
Rules:
active(
f(
f(
a))) →
mark(
f(
g(
f(
a))))
active(
g(
X)) →
g(
active(
X))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
a) →
ok(
a)
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
g(
ok(
X)) →
ok(
g(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:mark:ok → a:mark:ok
f :: a:mark:ok → a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok → a:mark:ok
g :: a:mark:ok → a:mark:ok
proper :: a:mark:ok → a:mark:ok
ok :: a:mark:ok → a:mark:ok
top :: a:mark:ok → top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat → a:mark:ok
Generator Equations:
gen_a:mark:ok3_0(0) ⇔ a
gen_a:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:mark:ok3_0(x))
The following defined symbols remain to be analysed:
f, active, g, proper, top
They will be analysed ascendingly in the following order:
f < active
g < active
active < top
f < proper
g < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol f.
(10) Obligation:
Innermost TRS:
Rules:
active(
f(
f(
a))) →
mark(
f(
g(
f(
a))))
active(
g(
X)) →
g(
active(
X))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
a) →
ok(
a)
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
g(
ok(
X)) →
ok(
g(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:mark:ok → a:mark:ok
f :: a:mark:ok → a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok → a:mark:ok
g :: a:mark:ok → a:mark:ok
proper :: a:mark:ok → a:mark:ok
ok :: a:mark:ok → a:mark:ok
top :: a:mark:ok → top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat → a:mark:ok
Generator Equations:
gen_a:mark:ok3_0(0) ⇔ a
gen_a:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:mark:ok3_0(x))
The following defined symbols remain to be analysed:
g, active, proper, top
They will be analysed ascendingly in the following order:
g < active
active < top
g < proper
proper < top
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol g.
(12) Obligation:
Innermost TRS:
Rules:
active(
f(
f(
a))) →
mark(
f(
g(
f(
a))))
active(
g(
X)) →
g(
active(
X))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
a) →
ok(
a)
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
g(
ok(
X)) →
ok(
g(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:mark:ok → a:mark:ok
f :: a:mark:ok → a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok → a:mark:ok
g :: a:mark:ok → a:mark:ok
proper :: a:mark:ok → a:mark:ok
ok :: a:mark:ok → a:mark:ok
top :: a:mark:ok → top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat → a:mark:ok
Generator Equations:
gen_a:mark:ok3_0(0) ⇔ a
gen_a:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:mark:ok3_0(x))
The following defined symbols remain to be analysed:
active, proper, top
They will be analysed ascendingly in the following order:
active < top
proper < top
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(14) Obligation:
Innermost TRS:
Rules:
active(
f(
f(
a))) →
mark(
f(
g(
f(
a))))
active(
g(
X)) →
g(
active(
X))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
a) →
ok(
a)
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
g(
ok(
X)) →
ok(
g(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:mark:ok → a:mark:ok
f :: a:mark:ok → a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok → a:mark:ok
g :: a:mark:ok → a:mark:ok
proper :: a:mark:ok → a:mark:ok
ok :: a:mark:ok → a:mark:ok
top :: a:mark:ok → top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat → a:mark:ok
Generator Equations:
gen_a:mark:ok3_0(0) ⇔ a
gen_a:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:mark:ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(16) Obligation:
Innermost TRS:
Rules:
active(
f(
f(
a))) →
mark(
f(
g(
f(
a))))
active(
g(
X)) →
g(
active(
X))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
a) →
ok(
a)
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
g(
ok(
X)) →
ok(
g(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:mark:ok → a:mark:ok
f :: a:mark:ok → a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok → a:mark:ok
g :: a:mark:ok → a:mark:ok
proper :: a:mark:ok → a:mark:ok
ok :: a:mark:ok → a:mark:ok
top :: a:mark:ok → top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat → a:mark:ok
Generator Equations:
gen_a:mark:ok3_0(0) ⇔ a
gen_a:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:mark:ok3_0(x))
The following defined symbols remain to be analysed:
top
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(18) Obligation:
Innermost TRS:
Rules:
active(
f(
f(
a))) →
mark(
f(
g(
f(
a))))
active(
g(
X)) →
g(
active(
X))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
a) →
ok(
a)
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
g(
ok(
X)) →
ok(
g(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:mark:ok → a:mark:ok
f :: a:mark:ok → a:mark:ok
a :: a:mark:ok
mark :: a:mark:ok → a:mark:ok
g :: a:mark:ok → a:mark:ok
proper :: a:mark:ok → a:mark:ok
ok :: a:mark:ok → a:mark:ok
top :: a:mark:ok → top
hole_a:mark:ok1_0 :: a:mark:ok
hole_top2_0 :: top
gen_a:mark:ok3_0 :: Nat → a:mark:ok
Generator Equations:
gen_a:mark:ok3_0(0) ⇔ a
gen_a:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:mark:ok3_0(x))
No more defined symbols left to analyse.