a__2nd(cons1(X, cons(Y, Z))) → mark(Y)
a__2nd(cons(X, X1)) → a__2nd(cons1(mark(X), mark(X1)))
a__from(X) → cons(mark(X), from(s(X)))
mark(2nd(X)) → a__2nd(mark(X))
mark(from(X)) → a__from(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(cons1(X1, X2)) → cons1(mark(X1), mark(X2))
a__2nd(X) → 2nd(X)
a__from(X) → from(X)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__2nd'(cons1'(X, cons'(Y, Z))) → mark'(Y)
a__2nd'(cons'(X, X1)) → a__2nd'(cons1'(mark'(X), mark'(X1)))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(cons1'(X1, X2)) → cons1'(mark'(X1), mark'(X2))
a__2nd'(X) → 2nd'(X)
a__from'(X) → from'(X)
Infered types.
Rules:
a__2nd'(cons1'(X, cons'(Y, Z))) → mark'(Y)
a__2nd'(cons'(X, X1)) → a__2nd'(cons1'(mark'(X), mark'(X1)))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(cons1'(X1, X2)) → cons1'(mark'(X1), mark'(X2))
a__2nd'(X) → 2nd'(X)
a__from'(X) → from'(X)
Types:
a__2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons1' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
mark' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
a__from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
s' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
_hole_cons':cons1':s':from':2nd'1 :: cons':cons1':s':from':2nd'
_gen_cons':cons1':s':from':2nd'2 :: Nat → cons':cons1':s':from':2nd'
Heuristically decided to analyse the following defined symbols:
a__2nd', mark', a__from'
They will be analysed ascendingly in the following order:
a__2nd' = mark'
a__2nd' = a__from'
mark' = a__from'
Rules:
a__2nd'(cons1'(X, cons'(Y, Z))) → mark'(Y)
a__2nd'(cons'(X, X1)) → a__2nd'(cons1'(mark'(X), mark'(X1)))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(cons1'(X1, X2)) → cons1'(mark'(X1), mark'(X2))
a__2nd'(X) → 2nd'(X)
a__from'(X) → from'(X)
Types:
a__2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons1' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
mark' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
a__from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
s' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
_hole_cons':cons1':s':from':2nd'1 :: cons':cons1':s':from':2nd'
_gen_cons':cons1':s':from':2nd'2 :: Nat → cons':cons1':s':from':2nd'
Generator Equations:
_gen_cons':cons1':s':from':2nd'2(0) ⇔ _hole_cons':cons1':s':from':2nd'1
_gen_cons':cons1':s':from':2nd'2(+(x, 1)) ⇔ cons1'(_hole_cons':cons1':s':from':2nd'1, _gen_cons':cons1':s':from':2nd'2(x))
The following defined symbols remain to be analysed:
mark', a__2nd', a__from'
They will be analysed ascendingly in the following order:
a__2nd' = mark'
a__2nd' = a__from'
mark' = a__from'
Proved the following rewrite lemma:
mark'(_gen_cons':cons1':s':from':2nd'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Induction Base:
mark'(_gen_cons':cons1':s':from':2nd'2(+(1, 0)))
Induction Step:
mark'(_gen_cons':cons1':s':from':2nd'2(+(1, +(_$n5, 1)))) →RΩ(1)
cons1'(mark'(_hole_cons':cons1':s':from':2nd'1), mark'(_gen_cons':cons1':s':from':2nd'2(+(1, _$n5)))) →IH
cons1'(mark'(_hole_cons':cons1':s':from':2nd'1), _*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
a__2nd'(cons1'(X, cons'(Y, Z))) → mark'(Y)
a__2nd'(cons'(X, X1)) → a__2nd'(cons1'(mark'(X), mark'(X1)))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(cons1'(X1, X2)) → cons1'(mark'(X1), mark'(X2))
a__2nd'(X) → 2nd'(X)
a__from'(X) → from'(X)
Types:
a__2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons1' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
mark' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
a__from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
s' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
_hole_cons':cons1':s':from':2nd'1 :: cons':cons1':s':from':2nd'
_gen_cons':cons1':s':from':2nd'2 :: Nat → cons':cons1':s':from':2nd'
Lemmas:
mark'(_gen_cons':cons1':s':from':2nd'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_cons':cons1':s':from':2nd'2(0) ⇔ _hole_cons':cons1':s':from':2nd'1
_gen_cons':cons1':s':from':2nd'2(+(x, 1)) ⇔ cons1'(_hole_cons':cons1':s':from':2nd'1, _gen_cons':cons1':s':from':2nd'2(x))
The following defined symbols remain to be analysed:
a__2nd', a__from'
They will be analysed ascendingly in the following order:
a__2nd' = mark'
a__2nd' = a__from'
mark' = a__from'
Could not prove a rewrite lemma for the defined symbol a__2nd'.
Rules:
a__2nd'(cons1'(X, cons'(Y, Z))) → mark'(Y)
a__2nd'(cons'(X, X1)) → a__2nd'(cons1'(mark'(X), mark'(X1)))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(cons1'(X1, X2)) → cons1'(mark'(X1), mark'(X2))
a__2nd'(X) → 2nd'(X)
a__from'(X) → from'(X)
Types:
a__2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons1' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
mark' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
a__from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
s' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
_hole_cons':cons1':s':from':2nd'1 :: cons':cons1':s':from':2nd'
_gen_cons':cons1':s':from':2nd'2 :: Nat → cons':cons1':s':from':2nd'
Lemmas:
mark'(_gen_cons':cons1':s':from':2nd'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_cons':cons1':s':from':2nd'2(0) ⇔ _hole_cons':cons1':s':from':2nd'1
_gen_cons':cons1':s':from':2nd'2(+(x, 1)) ⇔ cons1'(_hole_cons':cons1':s':from':2nd'1, _gen_cons':cons1':s':from':2nd'2(x))
The following defined symbols remain to be analysed:
a__from'
They will be analysed ascendingly in the following order:
a__2nd' = mark'
a__2nd' = a__from'
mark' = a__from'
Could not prove a rewrite lemma for the defined symbol a__from'.
Rules:
a__2nd'(cons1'(X, cons'(Y, Z))) → mark'(Y)
a__2nd'(cons'(X, X1)) → a__2nd'(cons1'(mark'(X), mark'(X1)))
a__from'(X) → cons'(mark'(X), from'(s'(X)))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(from'(X)) → a__from'(mark'(X))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(cons1'(X1, X2)) → cons1'(mark'(X1), mark'(X2))
a__2nd'(X) → 2nd'(X)
a__from'(X) → from'(X)
Types:
a__2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons1' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
cons' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
mark' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
a__from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
from' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
s' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
2nd' :: cons':cons1':s':from':2nd' → cons':cons1':s':from':2nd'
_hole_cons':cons1':s':from':2nd'1 :: cons':cons1':s':from':2nd'
_gen_cons':cons1':s':from':2nd'2 :: Nat → cons':cons1':s':from':2nd'
Lemmas:
mark'(_gen_cons':cons1':s':from':2nd'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_cons':cons1':s':from':2nd'2(0) ⇔ _hole_cons':cons1':s':from':2nd'1
_gen_cons':cons1':s':from':2nd'2(+(x, 1)) ⇔ cons1'(_hole_cons':cons1':s':from':2nd'1, _gen_cons':cons1':s':from':2nd'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
mark'(_gen_cons':cons1':s':from':2nd'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)