a__zeros → cons(0, zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeros → zeros
a__tail(X) → tail(X)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__zeros' → cons'(0', zeros')
a__tail'(cons'(X, XS)) → mark'(XS)
mark'(zeros') → a__zeros'
mark'(tail'(X)) → a__tail'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(0') → 0'
a__zeros' → zeros'
a__tail'(X) → tail'(X)
Infered types.
Rules:
a__zeros' → cons'(0', zeros')
a__tail'(cons'(X, XS)) → mark'(XS)
mark'(zeros') → a__zeros'
mark'(tail'(X)) → a__tail'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(0') → 0'
a__zeros' → zeros'
a__tail'(X) → tail'(X)
Types:
a__zeros' :: 0':zeros':cons':tail'
cons' :: 0':zeros':cons':tail' → 0':zeros':cons':tail' → 0':zeros':cons':tail'
0' :: 0':zeros':cons':tail'
zeros' :: 0':zeros':cons':tail'
a__tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
mark' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
_hole_0':zeros':cons':tail'1 :: 0':zeros':cons':tail'
_gen_0':zeros':cons':tail'2 :: Nat → 0':zeros':cons':tail'
Heuristically decided to analyse the following defined symbols:
a__tail', mark'
They will be analysed ascendingly in the following order:
a__tail' = mark'
Rules:
a__zeros' → cons'(0', zeros')
a__tail'(cons'(X, XS)) → mark'(XS)
mark'(zeros') → a__zeros'
mark'(tail'(X)) → a__tail'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(0') → 0'
a__zeros' → zeros'
a__tail'(X) → tail'(X)
Types:
a__zeros' :: 0':zeros':cons':tail'
cons' :: 0':zeros':cons':tail' → 0':zeros':cons':tail' → 0':zeros':cons':tail'
0' :: 0':zeros':cons':tail'
zeros' :: 0':zeros':cons':tail'
a__tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
mark' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
_hole_0':zeros':cons':tail'1 :: 0':zeros':cons':tail'
_gen_0':zeros':cons':tail'2 :: Nat → 0':zeros':cons':tail'
Generator Equations:
_gen_0':zeros':cons':tail'2(0) ⇔ 0'
_gen_0':zeros':cons':tail'2(+(x, 1)) ⇔ cons'(_gen_0':zeros':cons':tail'2(x), 0')
The following defined symbols remain to be analysed:
mark', a__tail'
They will be analysed ascendingly in the following order:
a__tail' = mark'
Proved the following rewrite lemma:
mark'(_gen_0':zeros':cons':tail'2(_n4)) → _gen_0':zeros':cons':tail'2(_n4), rt ∈ Ω(1 + n4)
Induction Base:
mark'(_gen_0':zeros':cons':tail'2(0)) →RΩ(1)
0'
Induction Step:
mark'(_gen_0':zeros':cons':tail'2(+(_$n5, 1))) →RΩ(1)
cons'(mark'(_gen_0':zeros':cons':tail'2(_$n5)), 0') →IH
cons'(_gen_0':zeros':cons':tail'2(_$n5), 0')
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
a__zeros' → cons'(0', zeros')
a__tail'(cons'(X, XS)) → mark'(XS)
mark'(zeros') → a__zeros'
mark'(tail'(X)) → a__tail'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(0') → 0'
a__zeros' → zeros'
a__tail'(X) → tail'(X)
Types:
a__zeros' :: 0':zeros':cons':tail'
cons' :: 0':zeros':cons':tail' → 0':zeros':cons':tail' → 0':zeros':cons':tail'
0' :: 0':zeros':cons':tail'
zeros' :: 0':zeros':cons':tail'
a__tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
mark' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
_hole_0':zeros':cons':tail'1 :: 0':zeros':cons':tail'
_gen_0':zeros':cons':tail'2 :: Nat → 0':zeros':cons':tail'
Lemmas:
mark'(_gen_0':zeros':cons':tail'2(_n4)) → _gen_0':zeros':cons':tail'2(_n4), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_0':zeros':cons':tail'2(0) ⇔ 0'
_gen_0':zeros':cons':tail'2(+(x, 1)) ⇔ cons'(_gen_0':zeros':cons':tail'2(x), 0')
The following defined symbols remain to be analysed:
a__tail'
They will be analysed ascendingly in the following order:
a__tail' = mark'
Could not prove a rewrite lemma for the defined symbol a__tail'.
Rules:
a__zeros' → cons'(0', zeros')
a__tail'(cons'(X, XS)) → mark'(XS)
mark'(zeros') → a__zeros'
mark'(tail'(X)) → a__tail'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(0') → 0'
a__zeros' → zeros'
a__tail'(X) → tail'(X)
Types:
a__zeros' :: 0':zeros':cons':tail'
cons' :: 0':zeros':cons':tail' → 0':zeros':cons':tail' → 0':zeros':cons':tail'
0' :: 0':zeros':cons':tail'
zeros' :: 0':zeros':cons':tail'
a__tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
mark' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
tail' :: 0':zeros':cons':tail' → 0':zeros':cons':tail'
_hole_0':zeros':cons':tail'1 :: 0':zeros':cons':tail'
_gen_0':zeros':cons':tail'2 :: Nat → 0':zeros':cons':tail'
Lemmas:
mark'(_gen_0':zeros':cons':tail'2(_n4)) → _gen_0':zeros':cons':tail'2(_n4), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_0':zeros':cons':tail'2(0) ⇔ 0'
_gen_0':zeros':cons':tail'2(+(x, 1)) ⇔ cons'(_gen_0':zeros':cons':tail'2(x), 0')
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_0':zeros':cons':tail'2(_n4)) → _gen_0':zeros':cons':tail'2(_n4), rt ∈ Ω(1 + n4)