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))
Renamed function symbols to avoid clashes with predefined symbol.
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))
Infered types.
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':ok'1 :: a':mark':ok'
_hole_top'2 :: top'
_gen_a':mark':ok'3 :: Nat → a':mark':ok'
Heuristically decided to analyse the following defined symbols:
active', f', g', proper', top'
They will be analysed ascendingly in the following order:
f' < active'
g' < active'
active' < top'
f' < proper'
g' < proper'
proper' < top'
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':ok'1 :: a':mark':ok'
_hole_top'2 :: top'
_gen_a':mark':ok'3 :: Nat → a':mark':ok'
Generator Equations:
_gen_a':mark':ok'3(0) ⇔ a'
_gen_a':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':mark':ok'3(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'
Could not prove a rewrite lemma for the defined symbol f'.
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':ok'1 :: a':mark':ok'
_hole_top'2 :: top'
_gen_a':mark':ok'3 :: Nat → a':mark':ok'
Generator Equations:
_gen_a':mark':ok'3(0) ⇔ a'
_gen_a':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':mark':ok'3(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'
Proved the following rewrite lemma:
g'(_gen_a':mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
Induction Base:
g'(_gen_a':mark':ok'3(+(1, 0)))
Induction Step:
g'(_gen_a':mark':ok'3(+(1, +(_$n12, 1)))) →RΩ(1)
mark'(g'(_gen_a':mark':ok'3(+(1, _$n12)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':ok'1 :: a':mark':ok'
_hole_top'2 :: top'
_gen_a':mark':ok'3 :: Nat → a':mark':ok'
Lemmas:
g'(_gen_a':mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
Generator Equations:
_gen_a':mark':ok'3(0) ⇔ a'
_gen_a':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':mark':ok'3(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'
Could not prove a rewrite lemma for the defined symbol active'.
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':ok'1 :: a':mark':ok'
_hole_top'2 :: top'
_gen_a':mark':ok'3 :: Nat → a':mark':ok'
Lemmas:
g'(_gen_a':mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
Generator Equations:
_gen_a':mark':ok'3(0) ⇔ a'
_gen_a':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':mark':ok'3(x))
The following defined symbols remain to be analysed:
proper', top'
They will be analysed ascendingly in the following order:
proper' < top'
Could not prove a rewrite lemma for the defined symbol proper'.
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':ok'1 :: a':mark':ok'
_hole_top'2 :: top'
_gen_a':mark':ok'3 :: Nat → a':mark':ok'
Lemmas:
g'(_gen_a':mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
Generator Equations:
_gen_a':mark':ok'3(0) ⇔ a'
_gen_a':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':mark':ok'3(x))
The following defined symbols remain to be analysed:
top'
Could not prove a rewrite lemma for the defined symbol top'.
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':ok'1 :: a':mark':ok'
_hole_top'2 :: top'
_gen_a':mark':ok'3 :: Nat → a':mark':ok'
Lemmas:
g'(_gen_a':mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
Generator Equations:
_gen_a':mark':ok'3(0) ⇔ a'
_gen_a':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_a':mark':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
g'(_gen_a':mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)