(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
from(X) → cons(X, from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)
Rewrite Strategy: FULL
(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
from(X) → cons(X, from(s(X)))
after(0', XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)
S is empty.
Rewrite Strategy: FULL
(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(4) Obligation:
TRS:
Rules:
from(X) → cons(X, from(s(X)))
after(0', XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)
Types:
from :: s:0' → cons
cons :: s:0' → cons → cons
s :: s:0' → s:0'
after :: s:0' → cons → cons
0' :: s:0'
hole_cons1_0 :: cons
hole_s:0'2_0 :: s:0'
gen_cons3_0 :: Nat → cons
gen_s:0'4_0 :: Nat → s:0'
(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
from, after
(6) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
from(
s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
X,
XS)) →
after(
N,
XS)
Types:
from :: s:0' → cons
cons :: s:0' → cons → cons
s :: s:0' → s:0'
after :: s:0' → cons → cons
0' :: s:0'
hole_cons1_0 :: cons
hole_s:0'2_0 :: s:0'
gen_cons3_0 :: Nat → cons
gen_s:0'4_0 :: Nat → s:0'
Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
The following defined symbols remain to be analysed:
from, after
(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol from.
(8) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
from(
s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
X,
XS)) →
after(
N,
XS)
Types:
from :: s:0' → cons
cons :: s:0' → cons → cons
s :: s:0' → s:0'
after :: s:0' → cons → cons
0' :: s:0'
hole_cons1_0 :: cons
hole_s:0'2_0 :: s:0'
gen_cons3_0 :: Nat → cons
gen_s:0'4_0 :: Nat → s:0'
Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
The following defined symbols remain to be analysed:
after
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
after(
gen_s:0'4_0(
n57_0),
gen_cons3_0(
n57_0)) →
gen_cons3_0(
0), rt ∈ Ω(1 + n57
0)
Induction Base:
after(gen_s:0'4_0(0), gen_cons3_0(0)) →RΩ(1)
gen_cons3_0(0)
Induction Step:
after(gen_s:0'4_0(+(n57_0, 1)), gen_cons3_0(+(n57_0, 1))) →RΩ(1)
after(gen_s:0'4_0(n57_0), gen_cons3_0(n57_0)) →IH
gen_cons3_0(0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
from(
s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
X,
XS)) →
after(
N,
XS)
Types:
from :: s:0' → cons
cons :: s:0' → cons → cons
s :: s:0' → s:0'
after :: s:0' → cons → cons
0' :: s:0'
hole_cons1_0 :: cons
hole_s:0'2_0 :: s:0'
gen_cons3_0 :: Nat → cons
gen_s:0'4_0 :: Nat → s:0'
Lemmas:
after(gen_s:0'4_0(n57_0), gen_cons3_0(n57_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n570)
Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
No more defined symbols left to analyse.
(12) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
after(gen_s:0'4_0(n57_0), gen_cons3_0(n57_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n570)
(13) BOUNDS(n^1, INF)
(14) Obligation:
TRS:
Rules:
from(
X) →
cons(
X,
from(
s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
X,
XS)) →
after(
N,
XS)
Types:
from :: s:0' → cons
cons :: s:0' → cons → cons
s :: s:0' → s:0'
after :: s:0' → cons → cons
0' :: s:0'
hole_cons1_0 :: cons
hole_s:0'2_0 :: s:0'
gen_cons3_0 :: Nat → cons
gen_s:0'4_0 :: Nat → s:0'
Lemmas:
after(gen_s:0'4_0(n57_0), gen_cons3_0(n57_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n570)
Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
No more defined symbols left to analyse.
(15) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
after(gen_s:0'4_0(n57_0), gen_cons3_0(n57_0)) → gen_cons3_0(0), rt ∈ Ω(1 + n570)
(16) BOUNDS(n^1, INF)