active(f(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
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'(X, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(f'(X, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':b':c':ok' → mark':b':c':ok'
f' :: mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok'
g' :: mark':b':c':ok' → mark':b':c':ok'
mark' :: mark':b':c':ok' → mark':b':c':ok'
b' :: mark':b':c':ok'
c' :: mark':b':c':ok'
proper' :: mark':b':c':ok' → mark':b':c':ok'
ok' :: mark':b':c':ok' → mark':b':c':ok'
top' :: mark':b':c':ok' → top'
_hole_mark':b':c':ok'1 :: mark':b':c':ok'
_hole_top'2 :: top'
_gen_mark':b':c':ok'3 :: Nat → mark':b':c':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'(X, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':b':c':ok' → mark':b':c':ok'
f' :: mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok'
g' :: mark':b':c':ok' → mark':b':c':ok'
mark' :: mark':b':c':ok' → mark':b':c':ok'
b' :: mark':b':c':ok'
c' :: mark':b':c':ok'
proper' :: mark':b':c':ok' → mark':b':c':ok'
ok' :: mark':b':c':ok' → mark':b':c':ok'
top' :: mark':b':c':ok' → top'
_hole_mark':b':c':ok'1 :: mark':b':c':ok'
_hole_top'2 :: top'
_gen_mark':b':c':ok'3 :: Nat → mark':b':c':ok'
Generator Equations:
_gen_mark':b':c':ok'3(0) ⇔ b'
_gen_mark':b':c':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':b':c':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'(X, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':b':c':ok' → mark':b':c':ok'
f' :: mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok'
g' :: mark':b':c':ok' → mark':b':c':ok'
mark' :: mark':b':c':ok' → mark':b':c':ok'
b' :: mark':b':c':ok'
c' :: mark':b':c':ok'
proper' :: mark':b':c':ok' → mark':b':c':ok'
ok' :: mark':b':c':ok' → mark':b':c':ok'
top' :: mark':b':c':ok' → top'
_hole_mark':b':c':ok'1 :: mark':b':c':ok'
_hole_top'2 :: top'
_gen_mark':b':c':ok'3 :: Nat → mark':b':c':ok'
Generator Equations:
_gen_mark':b':c':ok'3(0) ⇔ b'
_gen_mark':b':c':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':b':c':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_mark':b':c':ok'3(+(1, _n27))) → _*4, rt ∈ Ω(n27)
Induction Base:
g'(_gen_mark':b':c':ok'3(+(1, 0)))
Induction Step:
g'(_gen_mark':b':c':ok'3(+(1, +(_$n28, 1)))) →RΩ(1)
mark'(g'(_gen_mark':b':c':ok'3(+(1, _$n28)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':b':c':ok' → mark':b':c':ok'
f' :: mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok'
g' :: mark':b':c':ok' → mark':b':c':ok'
mark' :: mark':b':c':ok' → mark':b':c':ok'
b' :: mark':b':c':ok'
c' :: mark':b':c':ok'
proper' :: mark':b':c':ok' → mark':b':c':ok'
ok' :: mark':b':c':ok' → mark':b':c':ok'
top' :: mark':b':c':ok' → top'
_hole_mark':b':c':ok'1 :: mark':b':c':ok'
_hole_top'2 :: top'
_gen_mark':b':c':ok'3 :: Nat → mark':b':c':ok'
Lemmas:
g'(_gen_mark':b':c':ok'3(+(1, _n27))) → _*4, rt ∈ Ω(n27)
Generator Equations:
_gen_mark':b':c':ok'3(0) ⇔ b'
_gen_mark':b':c':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':b':c':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, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':b':c':ok' → mark':b':c':ok'
f' :: mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok'
g' :: mark':b':c':ok' → mark':b':c':ok'
mark' :: mark':b':c':ok' → mark':b':c':ok'
b' :: mark':b':c':ok'
c' :: mark':b':c':ok'
proper' :: mark':b':c':ok' → mark':b':c':ok'
ok' :: mark':b':c':ok' → mark':b':c':ok'
top' :: mark':b':c':ok' → top'
_hole_mark':b':c':ok'1 :: mark':b':c':ok'
_hole_top'2 :: top'
_gen_mark':b':c':ok'3 :: Nat → mark':b':c':ok'
Lemmas:
g'(_gen_mark':b':c':ok'3(+(1, _n27))) → _*4, rt ∈ Ω(n27)
Generator Equations:
_gen_mark':b':c':ok'3(0) ⇔ b'
_gen_mark':b':c':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':b':c':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, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':b':c':ok' → mark':b':c':ok'
f' :: mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok'
g' :: mark':b':c':ok' → mark':b':c':ok'
mark' :: mark':b':c':ok' → mark':b':c':ok'
b' :: mark':b':c':ok'
c' :: mark':b':c':ok'
proper' :: mark':b':c':ok' → mark':b':c':ok'
ok' :: mark':b':c':ok' → mark':b':c':ok'
top' :: mark':b':c':ok' → top'
_hole_mark':b':c':ok'1 :: mark':b':c':ok'
_hole_top'2 :: top'
_gen_mark':b':c':ok'3 :: Nat → mark':b':c':ok'
Lemmas:
g'(_gen_mark':b':c':ok'3(+(1, _n27))) → _*4, rt ∈ Ω(n27)
Generator Equations:
_gen_mark':b':c':ok'3(0) ⇔ b'
_gen_mark':b':c':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':b':c':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, g'(X), Y)) → mark'(f'(Y, Y, Y))
active'(g'(b')) → mark'(c')
active'(b') → mark'(c')
active'(g'(X)) → g'(active'(X))
g'(mark'(X)) → mark'(g'(X))
proper'(f'(X1, X2, X3)) → f'(proper'(X1), proper'(X2), proper'(X3))
proper'(g'(X)) → g'(proper'(X))
proper'(b') → ok'(b')
proper'(c') → ok'(c')
f'(ok'(X1), ok'(X2), ok'(X3)) → ok'(f'(X1, X2, X3))
g'(ok'(X)) → ok'(g'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':b':c':ok' → mark':b':c':ok'
f' :: mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok' → mark':b':c':ok'
g' :: mark':b':c':ok' → mark':b':c':ok'
mark' :: mark':b':c':ok' → mark':b':c':ok'
b' :: mark':b':c':ok'
c' :: mark':b':c':ok'
proper' :: mark':b':c':ok' → mark':b':c':ok'
ok' :: mark':b':c':ok' → mark':b':c':ok'
top' :: mark':b':c':ok' → top'
_hole_mark':b':c':ok'1 :: mark':b':c':ok'
_hole_top'2 :: top'
_gen_mark':b':c':ok'3 :: Nat → mark':b':c':ok'
Lemmas:
g'(_gen_mark':b':c':ok'3(+(1, _n27))) → _*4, rt ∈ Ω(n27)
Generator Equations:
_gen_mark':b':c':ok'3(0) ⇔ b'
_gen_mark':b':c':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':b':c':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
g'(_gen_mark':b':c':ok'3(+(1, _n27))) → _*4, rt ∈ Ω(n27)