active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
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'(c') → mark'(f'(g'(c')))
active'(f'(g'(X))) → mark'(g'(X))
proper'(c') → ok'(c')
proper'(f'(X)) → f'(proper'(X))
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'(c') → mark'(f'(g'(c')))
active'(f'(g'(X))) → mark'(g'(X))
proper'(c') → ok'(c')
proper'(f'(X)) → f'(proper'(X))
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' :: c':mark':ok' → c':mark':ok'
c' :: c':mark':ok'
mark' :: c':mark':ok' → c':mark':ok'
f' :: c':mark':ok' → c':mark':ok'
g' :: c':mark':ok' → c':mark':ok'
proper' :: c':mark':ok' → c':mark':ok'
ok' :: c':mark':ok' → c':mark':ok'
top' :: c':mark':ok' → top'
_hole_c':mark':ok'1 :: c':mark':ok'
_hole_top'2 :: top'
_gen_c':mark':ok'3 :: Nat → c':mark':ok'
Heuristically decided to analyse the following defined symbols:
f', g', proper', top'
They will be analysed ascendingly in the following order:
f' < proper'
g' < proper'
proper' < top'
Rules:
active'(c') → mark'(f'(g'(c')))
active'(f'(g'(X))) → mark'(g'(X))
proper'(c') → ok'(c')
proper'(f'(X)) → f'(proper'(X))
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' :: c':mark':ok' → c':mark':ok'
c' :: c':mark':ok'
mark' :: c':mark':ok' → c':mark':ok'
f' :: c':mark':ok' → c':mark':ok'
g' :: c':mark':ok' → c':mark':ok'
proper' :: c':mark':ok' → c':mark':ok'
ok' :: c':mark':ok' → c':mark':ok'
top' :: c':mark':ok' → top'
_hole_c':mark':ok'1 :: c':mark':ok'
_hole_top'2 :: top'
_gen_c':mark':ok'3 :: Nat → c':mark':ok'
Generator Equations:
_gen_c':mark':ok'3(0) ⇔ c'
_gen_c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_c':mark':ok'3(x))
The following defined symbols remain to be analysed:
f', g', proper', top'
They will be analysed ascendingly in the following order:
f' < proper'
g' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
active'(c') → mark'(f'(g'(c')))
active'(f'(g'(X))) → mark'(g'(X))
proper'(c') → ok'(c')
proper'(f'(X)) → f'(proper'(X))
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' :: c':mark':ok' → c':mark':ok'
c' :: c':mark':ok'
mark' :: c':mark':ok' → c':mark':ok'
f' :: c':mark':ok' → c':mark':ok'
g' :: c':mark':ok' → c':mark':ok'
proper' :: c':mark':ok' → c':mark':ok'
ok' :: c':mark':ok' → c':mark':ok'
top' :: c':mark':ok' → top'
_hole_c':mark':ok'1 :: c':mark':ok'
_hole_top'2 :: top'
_gen_c':mark':ok'3 :: Nat → c':mark':ok'
Generator Equations:
_gen_c':mark':ok'3(0) ⇔ c'
_gen_c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_c':mark':ok'3(x))
The following defined symbols remain to be analysed:
g', proper', top'
They will be analysed ascendingly in the following order:
g' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol g'.
Rules:
active'(c') → mark'(f'(g'(c')))
active'(f'(g'(X))) → mark'(g'(X))
proper'(c') → ok'(c')
proper'(f'(X)) → f'(proper'(X))
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' :: c':mark':ok' → c':mark':ok'
c' :: c':mark':ok'
mark' :: c':mark':ok' → c':mark':ok'
f' :: c':mark':ok' → c':mark':ok'
g' :: c':mark':ok' → c':mark':ok'
proper' :: c':mark':ok' → c':mark':ok'
ok' :: c':mark':ok' → c':mark':ok'
top' :: c':mark':ok' → top'
_hole_c':mark':ok'1 :: c':mark':ok'
_hole_top'2 :: top'
_gen_c':mark':ok'3 :: Nat → c':mark':ok'
Generator Equations:
_gen_c':mark':ok'3(0) ⇔ c'
_gen_c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_c':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'(c') → mark'(f'(g'(c')))
active'(f'(g'(X))) → mark'(g'(X))
proper'(c') → ok'(c')
proper'(f'(X)) → f'(proper'(X))
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' :: c':mark':ok' → c':mark':ok'
c' :: c':mark':ok'
mark' :: c':mark':ok' → c':mark':ok'
f' :: c':mark':ok' → c':mark':ok'
g' :: c':mark':ok' → c':mark':ok'
proper' :: c':mark':ok' → c':mark':ok'
ok' :: c':mark':ok' → c':mark':ok'
top' :: c':mark':ok' → top'
_hole_c':mark':ok'1 :: c':mark':ok'
_hole_top'2 :: top'
_gen_c':mark':ok'3 :: Nat → c':mark':ok'
Generator Equations:
_gen_c':mark':ok'3(0) ⇔ c'
_gen_c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_c':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'(c') → mark'(f'(g'(c')))
active'(f'(g'(X))) → mark'(g'(X))
proper'(c') → ok'(c')
proper'(f'(X)) → f'(proper'(X))
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' :: c':mark':ok' → c':mark':ok'
c' :: c':mark':ok'
mark' :: c':mark':ok' → c':mark':ok'
f' :: c':mark':ok' → c':mark':ok'
g' :: c':mark':ok' → c':mark':ok'
proper' :: c':mark':ok' → c':mark':ok'
ok' :: c':mark':ok' → c':mark':ok'
top' :: c':mark':ok' → top'
_hole_c':mark':ok'1 :: c':mark':ok'
_hole_top'2 :: top'
_gen_c':mark':ok'3 :: Nat → c':mark':ok'
Generator Equations:
_gen_c':mark':ok'3(0) ⇔ c'
_gen_c':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_c':mark':ok'3(x))
No more defined symbols left to analyse.