active(f(X)) → mark(g(h(f(X))))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(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'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(f'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Heuristically decided to analyse the following defined symbols:
active', g', h', f', proper', top'
They will be analysed ascendingly in the following order:
g' < active'
h' < active'
f' < active'
active' < top'
g' < proper'
h' < proper'
f' < proper'
proper' < top'
Rules:
active'(f'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
g', active', h', f', proper', top'
They will be analysed ascendingly in the following order:
g' < active'
h' < active'
f' < active'
active' < top'
g' < proper'
h' < proper'
f' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol g'.
Rules:
active'(f'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
h', active', f', proper', top'
They will be analysed ascendingly in the following order:
h' < active'
f' < active'
active' < top'
h' < proper'
f' < proper'
proper' < top'
Proved the following rewrite lemma:
h'(_gen_mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
Induction Base:
h'(_gen_mark':ok'3(+(1, 0)))
Induction Step:
h'(_gen_mark':ok'3(+(1, +(_$n12, 1)))) →RΩ(1)
mark'(h'(_gen_mark':ok'3(+(1, _$n12)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
h'(_gen_mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
f', active', proper', top'
They will be analysed ascendingly in the following order:
f' < active'
active' < top'
f' < proper'
proper' < top'
Proved the following rewrite lemma:
f'(_gen_mark':ok'3(+(1, _n592))) → _*4, rt ∈ Ω(n592)
Induction Base:
f'(_gen_mark':ok'3(+(1, 0)))
Induction Step:
f'(_gen_mark':ok'3(+(1, +(_$n593, 1)))) →RΩ(1)
mark'(f'(_gen_mark':ok'3(+(1, _$n593)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
h'(_gen_mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
f'(_gen_mark':ok'3(+(1, _n592))) → _*4, rt ∈ Ω(n592)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_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'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
h'(_gen_mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
f'(_gen_mark':ok'3(+(1, _n592))) → _*4, rt ∈ Ω(n592)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_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'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
h'(_gen_mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
f'(_gen_mark':ok'3(+(1, _n592))) → _*4, rt ∈ Ω(n592)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_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'(X)) → mark'(g'(h'(f'(X))))
active'(f'(X)) → f'(active'(X))
active'(h'(X)) → h'(active'(X))
f'(mark'(X)) → mark'(f'(X))
h'(mark'(X)) → mark'(h'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(g'(X)) → g'(proper'(X))
proper'(h'(X)) → h'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
g'(ok'(X)) → ok'(g'(X))
h'(ok'(X)) → ok'(h'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
f' :: mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
g' :: mark':ok' → mark':ok'
h' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
h'(_gen_mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)
f'(_gen_mark':ok'3(+(1, _n592))) → _*4, rt ∈ Ω(n592)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
h'(_gen_mark':ok'3(+(1, _n11))) → _*4, rt ∈ Ω(n11)