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